Pierre Zakrzewski was working with correspondent Benjamin Hall when their vehicle was hit in the north-eastern part of the village of Gorenka, near Kyiv.
Fox News announced the attack and Mr Hall’s injury on Monday by saying the reporter “was injured while newsgathering outside of Kyiv in Ukraine”.
Ukrainian journalist Oleksandra Kuvshinova was also reported to be killed in the attack.
Fox News presenter Bill Hemmer said: “Pierre Zakrzewski was an absolute legend at this network and his loss is devastating.
“He has been with us for years, covering wars in Iraq, Afghanistan and Syria. Our CEO Susanne Scott noted a few moments ago that Pierre jumped in to help out with all sorts of roles in the field – photographer and engineer and editor and producer and he did it all under immense pressure and with tremendous skill.”
Fox News producer Yonat Friling paid tribute as she wrote on Twitter: “In yesterday’s attack near Kyiv, we have lost a beautiful brave woman – Oleksandra Kuvshinova – Sasha.
“She loved music and she was funny and kind. She was 24-years-old. She worked with our team for the past month and did a brilliant job. May her memory be a blessing.”
Ukraine news website kp.ua reported Mr Zakrzewski and Ms Kuvshinova died “as a result of artillery shelling by Russian troops in the north-eastern part of the village of Gorenka”, although more details were not given.
Irish Prime Minister Micheal Martin paid tribute to Mr Zakrzewski, who was also Irish, as he said: “Deeply disturbed and saddened by the killing of Irish citizen and journalist Pierre Zakrzewski and one of his colleagues today.
“My thoughts are with their families, friends and fellow journalists.
“We condemn this indiscriminate and immoral war by Russia on Ukraine.”
Irish President Michael D Higgins said in a statement: “The indiscriminate killing of civilians, including journalists, must be brought to an end.”
Fox News anchor John Roberts said it was “horrible news to report”.
He wrote on Twitter: “I worked with Pierre many times around the world. He was an absolute treasure. Sending our most heartfelt prayers to Pierre’s wife and family.”
It comes days after US filmmaker Brent Renaud was killed in Ukraine after Russian forces opened fire on his vehicle.
The 50-year-old, from Arkansas, was gathering material for a report about refugees when his vehicle was hit at a checkpoint in Irpin, just outside Kyiv.
Fox Newslater released an updated statement to confirm Ms Kuvshynova’s death and said she had been working as a consultant for the channel.
It read: “We held off on delivering this devastating news earlier today out of respect for her family whom we have been in touch with throughout.”