Biden on Tuesday told reporters he had spoken with McConnell and would meet with him soon.

On Tuesday, McConnell said Biden has "devoted himself to public service for many years" and congratulated Sen. Kamala Harris (D-Calif.) for becoming the first woman elected vice president. Harris later told ABC News that she applauded McConnell for speaking with Biden.

But McConnell was careful not to begin his comments Tuesday by congratulating Biden after Trump's defeat in the Electoral College on Monday and in the Supreme Court last week. Trump has continued tweeting angrily about the election, falsely claiming on Tuesday that there are "tremendous problems being found with voting machines."

“President Trump should take his cue from Leader McConnell that it’s time to end his term with a modicum of grace and dignity,” said Senate Minority Leader Chuck Schumer.

McConnell's recognition of Biden came at the tail end of a long speech praising Trump's accomplishments and after Senate Chaplain Barry Black recognized Biden and Harris as victors. McConnell called Trump an "outsider who swore he would shake up Washington and lead our country to new accomplishments both at home and abroad" and did "exactly that."

McConnell had carefully tiptoed around Biden's electoral win as Trump fought the election in the courts. But the president has been handed a series of losses, punctuated by the Electoral College clinching Biden's win. That prompted a flood of Republicans to recognize him, including the top Republican in Congress.

"As of this morning, our country has officially a president-elect and a vice president-elect. Many of us hoped that the presidential election would yield a different result. But our system of government has processes to determine who will be sworn in on Jan. 20," McConnell said.