The President said the federal agency overseeing the handover must “do what needs to be done”, even as he vowed to keep contesting his election defeat.
The General Services Administration said it was acknowledging Mr Biden as the “apparent winner”.
It came as Biden’s victory in the state of Michigan was officially certified, a major blow to Trump.
After the GSA announcement, the Pentagon said it would provide “support to the Biden team… in a professional, orderly, and efficient manner that is befitting of the public’s expectation of the Department and our commitment to national security”.
The Biden team welcomed the start of the transition process as the Democratic president-elect gears up to be sworn in on January 20.
“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,” a team statement said.
Earlier, Biden unveiled a foreign policy and national security team consisting of old colleagues from his years in the Obama administration.
He will appoint Anthony Blinken as Secretary of State and John Kerry as climate envoy, while Janet Yellen is tipped to be the first female US Treasury Secretary.
Trump tweeted as the GSA, which is tasked with formally beginning presidential transitions, informed the Biden camp that it would start the transition process.
Administrator, Emily Murphy, said she was making $6.3m in funds available to the president-elect.
While pledging to keep up the “good fight”, the President 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.”
Trump’s fellow Republicans have increasingly been breaking ranks over the transition, with several of them speaking out on Monday.
Tennessee Senator, Lamar Alexander, who is retiring, released a statement saying that Trump should “put the country first” and help Biden succeed.
“When you are in public life, people remember the last thing you do,” Alexander said.