Joe Biden wearing a suit and tie© Alex Wong/Getty ImagesWASHINGTON—President-elect Joe Biden is under pressure from several groups within the Democratic Party as he weighs more cabinet picks, with Black and Latino lawmakers discussing coordinating efforts to try to sway his selections for top jobs.

Members of the Congressional Black Caucus and Congressional Hispanic Caucus have been in touch with Mr. Biden’s transition team to encourage him to increase diversity at the highest levels of the government. Some of those members are now discussing tactics to elevate their preferred contenders, including possibly aligning their endorsements. They haven’t reached an agreement on their strategy, according to people familiar with the discussions, but one approach would be to choose one Black or Latino candidate to recommend for a handful of top remaining positions.

“The African-American and Latino communities rallied behind this administration and their expectations are to be included at all levels,” said Rep. Bennie Thompson (D, Miss.), who has spoken with the Biden transition team.

The efforts underscore the pressures Mr. Biden faces as he builds out his administration. In addition to demands that he increase the diversity across his cabinet nominees, Mr. Biden is navigating the party’s progressive base and his pledge to govern as a consensus builder. Some of the candidates being pushed by Black and Latino groups don’t have the backing of progressive groups, and vice versa.

Mr. Biden’s team so far has included a number of firsts for women and minorities. They include his choice of Alejandro Mayorkas, who is Cuban-American, to be the first Latino and first immigrant to lead the Department of Homeland Security, and Cecilia Rouse, who would be the first Black chair of the Council of Economic Advisers. Mr. Biden named another Black woman, Linda Thomas-Greenfield, as his nominee for ambassador to the United Nations, and has appointed more than a dozen people of color to senior White House roles.

“President-elect Biden is working to build an administration that looks like America,” transition spokesman Jamal Brown said in a statement. “Our team is engaging leaders and organizations to ensure they have a seat at the table in helping to develop and implement the President-elect’s vision.”

Mr. Biden’s team has been tight-lipped so far about how he is picking people to join his administration. Mr. Brown said meetings with outside groups and leaders would continue over the coming weeks. The transition has also engaged with civil-rights groups, such as the NAACP Legal Defense Fund and the National Urban League.

Mr. Biden has filled two top cabinet positions with Antony Blinken at the State Department and Janet Yellen at the Treasury Department. Both are white; Ms. Yellen would be the first woman to serve in that position. The roles of attorney general and defense secretary, the other positions considered to be in the top four, haven’t been filled.

So far, Black and Hispanic lawmakers have engaged in separate talks with the Biden transition team to convey their recommendations.

Mr. Biden’s aides will meet virtually with members of the Hispanic caucus on Thursday, according to a person close to the transition. Mr. Biden’s incoming chief of staff, Ron Klain, and transition co-chairs Jeff Zients and Ted Kaufman are expected to attend, according to a briefing document viewed by The Wall Street Journal. Members of Mr. Biden’s transition team have also held conversations with members of the CBC.

Lawmakers on the call Thursday plan to repeat their request that the Biden team ensure that Latinos account for 20% of the federal workforce and that five Latinos are appointed to the cabinet, with at least one in a top position, according to the document.

Members of the Hispanic caucus called on Mr. Biden in a letter to nominate either Democratic National Committee Chairman Tom Perez, who was labor secretary in the Obama administration, or California Attorney General Xavier Becerra to lead the Justice Department.

“There are a lot of things that he needs to pay attention to, that the administration needs to focus on, because some Latinos in the country are feeling ignored by the Democratic Party,” said Rep. Vicente Gonzalez (D., Texas). ”They clearly know and understand our urgency. We’re hoping to see results.”

Mr. Biden has considered several people for the attorney general job, including former Massachusetts Gov. Deval Patrick, who is Black.

The CHC in a letter Sunday recommended Mr. Biden pick New Mexico Gov. Michelle Lujan Grisham to lead the Department of Health and Human Services. She was offered the role of interior secretary but turned it down, a person close to Mr. Biden’s transition said. She remains under consideration for HHS secretary but, according to people familiar with the process, faces competition from Rhode Island Gov. Gina Raimondo. Neither Ms. Lujan Grisham nor Ms. Raimondo immediately responded to requests for comment.

Members of the CBC have encouraged Mr. Biden to choose a Black nominee for defense secretary. At least two Black candidates, retired Army General Lloyd Austin—who ran the military’s Central Command during the Obama years—and former DHS Secretary Jeh Johnson, are under consideration.

Michèle Flournoy, a former Defense Department official under Presidents Barack Obama and Bill Clinton, had been seen by many defense experts as a favorite for that job. She has come under scrutiny from some in the more liberal wing of the Democratic Party because of her role as a board member of Booz Allen Hamilton Holding Corp., a Pentagon contractor, and her support of the surge of troops in Afghanistan during the Obama administration.

On Wednesday, roughly 100 people held a call focused on advocating for progressive foreign policy personnel in a Biden administration. Yasmine Taeb, a senior fellow at the progressive think tank Center for International Policy, who was leading the call, said even though they are unhappy with parts of her record, she and other progressives had been advised by some lawmakers not to advocate against Ms. Flournoy because they prefer her to Mr. Johnson. Some advocates of more women in military positions have also backed Ms. Flournoy.

Ms. Taeb said after the call that she and others were concerned about Mr. Johnson’s role in drone policy while he was general counsel at the Defense Department. The Obama administration was criticized for civilian casualties as a result of its use of drone strikes.

A person close to Ms. Flournoy said her work in the private sector doesn’t influence her policy views. “In fact, her cumulative experiences would make her better able to oversee the defense industry,” the person said.

Mr. Johnson hasn’t commented about any potential role in the administration.

Jeff Hauser, who runs the Revolving Door Project, a progressive group focused on executive branch appointments, described Mr. Biden’s cabinet so far as “not fantastic, but noticeably better than in 2009” under Mr. Obama.

Progressive organizations have sounded the alarm about former Chicago Mayor Rahm Emanuel, who is among those being considered to lead the Transportation Department. Some in the party view Mr. Emanuel, a close Biden ally and former chief of staff to Mr. Obama, as too pro-business, and are critical of his handling as Chicago mayor of a 2014 police shooting of a Black teenager.

Mr. Emanuel didn’t respond to a request for comment.

Some also were critical of Mr. Biden’s decision to nominate Neera Tanden, who has tussled on Twitter with supporters of Mr. Sanders and other progressives, to lead the Office of Management and Budget. Ms. Tanden, who if confirmed would be the first Indian-American to lead OMB, has also faced pushback from Republicans. Mr. Biden told a New York Times columnist that Ms. Tanden was “smart as hell” and that Republicans were going to criticize some of his nominees no matter who he picked.

Aides to Sen. Bernie Sanders of Vermont, who ran against Mr. Biden in the Democratic primary, have been in conversation with Mr. Biden’s team about his interest in becoming labor secretary. Mr. Biden said last week on NBC that nothing was off the table but that it would be difficult to take lawmakers out of Congress where margins are tight. House Democrats’ will have a narrow majority in their chamber and the thin Senate majority will be determined by early January runoffs in Georgia.

Our Revolution, a group aligned with Mr. Sanders, pushed in an email to supporters for Rep. Debra Haaland of New Mexico to be interior secretary. The group lobbied the transition in a letter Tuesday for Rep. Marcia Fudge of Ohio to lead the Agriculture Department. Some Black leaders have also promoted Ms. Fudge.

Several civil rights organizations—including the NAACP, the National Urban League, and the National Action Network—are also seeking to speak with Mr. Biden directly.

“It’s a gentle but firm push to do more,” said Rev. Al Sharpton, founder of the National Action Network.

“We don’t want us to be seated at the dining room table when everyone else has finished the meal,” he added.