The General Services Administration chief Emily Murphy, who had been formally holding up the transition for more than two weeks citing uncertainty in the results, said Monday that her agency would provide Mr. Biden federal resources meant to assist in a smooth transfer of power.

Mr. Trump said he had instructed aides to cooperate with the transition approved by the GSA, though he pledged to continue his legal efforts to overturn the election result.

“I believe we will prevail!” he said. “Nevertheless, in the best interest of our Country, I am recommending that Emily and her team do what needs to be done with regard to initial protocols, and have told my team to do the same,” he said.

Michigan on Monday certified the results of the election, and legal setbacks piled up for Mr. Trump and his allies. No evidence of significant voter fraud has been produced, and Republican allies in Congress had signaled growing impatience with Mr. Trump and his team, though party leaders haven’t publicly pushed him to concede.

Mr. Trump has given no indication that he plans to give any kind of formal concession speech, but he has been polling advisers on what his options are for an exit strategy, according to administration officials.

Over the last week, according to one of the officials, the president has appeared increasingly resigned to the reality that he will no longer be in the White House past Jan. 20, asking one adviser in a recent conversation: “What do you think I should do next?”

Some advisers have urged him to talk up another presidential run in 2024, even if he doesn’t plan to follow through. The president also has privately indicated he wants to keep up the appearance of a fight until the bitter end, telling one adviser: “Everybody loves a fighter.”

In a letter to Mr. Biden, Ms. Murphy, the GSA administrator, cited recent legal developments and certifications of state election results in explaining her decision to stop blocking the resources.

“I have determined that you may access the postelection resources and services described” in presidential transition legislation, she wrote. “The actual winner of the presidential election will be determined by the electoral process detailed in the Constitution.”

Ms. Murphy had declined to name the apparent winner of the election, a technical designation that allows the federal government to begin cooperating with the president-elect, in the aftermath of the Associated Press’s Nov. 7 decision to declare Mr. Biden the winner of the election.

Without the GSA designation, formally known as an ascertainment, Mr. Biden’s team wasn’t allowed to receive high-level classified briefings and was denied more than $6 million in government funds. Mr. Biden’s transition operation also had been blocked from accessing federal agencies, where career government officials had prepared briefing books to ease the transition from one administration to the next.

“Today’s decision is a needed step to begin tackling the challenges facing our nation, including getting the pandemic under control and our economy back on track,” said Yohannes Abraham, the executive director of Mr. Biden’s transition team, adding that the transition would begin meeting in the coming days with officials from federal agencies.

Lawmakers from both parties have worried that the GSA delay was beginning to hinder Mr. Biden’s transition, preventing him from receiving classified briefings and preparing for the distribution of a coronavirus vaccine expected to largely take place during his term in office. Presidents-elect typically get access to the president’s daily brief, which details the government’s most secret intelligence.

“This is probably the closest thing to a concession that President Trump could issue,” Senate Minority Leader Chuck Schumer (D., N.Y.) said in a statement Monday night. “Let us all now—Democrats and Republicans, the Trump Administration and the incoming Biden Administration—unite together for a smooth and peaceful transition that will benefit America.”

Ms. Murphy, in her letter, said her decision was “not made out of fear or favoritism,” adding that she hadn’t received any direction or pressure from Mr. Trump or administration officials to delay her decision. She said she had received threats meant “to coerce me into making this determination prematurely.”

Mr. Trump thanked Ms. Murphy, saying she had been “harassed, threatened, and abused—and I do not want to see this happen to her, her family, or employees of GSA. “

The House has received the letter from Ms. Murphy, Drew Hammill, spokesman for House Speaker Nancy Pelosi (D., Calif.), said Monday.

Senior Democrats in both the House and Senate had ramped up pressure on Ms. Murphy on Monday to make the official election ascertainment.

While GOP leaders haven’t said Mr. Trump should concede, a slowly expanding group of Republicans had said Mr. Biden’s transition should begin immediately.

Mr. Biden and Vice President-elect Kamala Harris “should begin receiving all appropriate briefings related to national security and Covid-19 to facilitate a smooth transfer of power in the likely event that they are to take office on Jan. 20,” Sen. Shelley Moore Capito (R., W.Va.) said Monday. While Mr. Trump has a right to request recounts and raise legal claims, “at some point, the 2020 election must end,” she said.

Sen. Rob Portman (R., Ohio) said in an opinion article in the Cincinnati Enquirer that the GSA should begin the transition process. “This is only prudent,” he wrote. “There is no evidence as of now of any widespread fraud or irregularities that would change the result in any state,” he said.

Mr. Trump should “put the country first and have a prompt and orderly transition to help the new administration succeed,” said Sen. Lamar Alexander (R., Tenn.), who is retiring. “When you are in public life, people remember the last thing you do.”

The mechanics of presidential transitions are governed by a 1963 federal law that leaves it to the administrator of the GSA to certify a winner and set the process in motion. Historically, the agency has accepted media projections of the winner as the basis for beginning the transition process.

However, the law is vague on what standard the GSA administrator should apply in making that determination—a potential barrier to any legal challenge by Mr. Biden. The law says only that the administration shall determine the “apparent” winner, with no instructions on what that should mean in the event of a dispute or a challenge. In past elections, with the exception of the 2000 election, the GSA has ascertained a winner within hours or days of election night.

WASHINGTON—President Trump said his administration would cooperate with President-elect Joe Biden’s transition to the White House, ending a delay that had come under increasing criticism from members of both parties as Mr. Trump’s long-shot effort to overturn the election foundered.