McConnell congratulated Biden and Vice President-elect Kamala Harris on their victory during remarks on the Senate floor on Tuesday, a day after the Electoral College formalized the Democrats' November 3 win.
The senator was among many congressional Republicans who had not acknowledged Biden's defeat of President Donald Trump, who has continued to make unfounded claims of election fraud and refused to concede.
Biden told reporters he spoke with McConnell before heading to Georgia, where he will campaign for two Democratic US Senate candidates whose January 5 runoff elections could make or break his domestic policy agenda.
"While we disagree on a lot of things, there are things we can work together on," Biden told reporters. "We agreed to get together sooner (rather) than later."
Hours after their call, McConnell told reporters that lawmakers would not leave Washington this year until they have agreed on a new coronavirus economic relief package. Top US congressional leaders are meeting on Tuesday as they seek to finalize a government funding bill and end a standoff on economic relief.
McConnell and his top deputies pressed other Senate Republicans not to join with any House Republicans who may object to the election outcome when Congress meets on January 6 to ratify the decision.
Any such effort to challenge the results is almost certain to fail since it would require approval by both chambers, including the Democratic-controlled House of Representatives.
Also on Tuesday, Biden was congratulated by Russian President Vladimir Putin, a favourite of Trump's, and Mexican President Andrés Manuel López Obrador.
Biden and Harris will be sworn in on the steps of the US Capitol on January 20 with far fewer people present than is customary due to the raging coronavirus that has killed more than 300,000 people in the United States, the inaugural planning committee said on Tuesday.
Festivities will be largely virtual, and the committee urged members of the public to refrain from traveling to Washington.
BIDEN CABINET PICKS
Biden has pressed ahead with building his Cabinet, with word of more picks emerging on Tuesday.
He has chosen Pete Buttigieg, his former rival for the Democratic presidential nomination, to lead the US Transportation Department, according to three people familiar with the matter. Buttigieg (38), the former mayor of South Bend, Indiana, would be the first LGBTQ person nominated to Biden's administration and, if confirmed, the first LGBTQ Cabinet secretary approved by the Senate.
US Representative Deb Haaland of New Mexico is Biden's leading choice to head the Interior Department, according to three sources familiar with the proceedings, a selection that would make her the first Native American to lead a Cabinet agency.
Biden's trip to Atlanta comes nine days after Trump travelled to Georgia in support of the Republican senators seeking re-election.
Biden's narrow win in Georgia last month underscored the Southern state's transformation from Republican stronghold to one of the country's most competitive political battlegrounds. That was illustrated again with more than 480,000 ballots cast after the first full day of in-person early voting and mail voting, according to the US Elections Project at the University of Florida.
Republican Senators David Perdue and Kelly Loeffler are facing Democratic challengers Jon Ossoff and Raphael Warnock, respectively, in twin races that will determine which party will control the US Senate.
If the Republicans win either contest, they would maintain power in the Senate, allowing them to thwart many of Biden's ambitious legislative goals on issues such as the coronavirus pandemic, the economy and climate change.
A Democratic sweep would give Biden's party control of the White House and both chambers of Congress. Democrats already hold a majority in the House of Representatives.
Trump campaign spokesman Tim Murtaugh said in a statement that Warnock and Ossoff "represent the left-most fringe" of the Democratic Party and that Biden campaigning for them showed he was under the influence of that wing of the party.
Biden said on Tuesday that he planned to get the coronavirus vaccine "sooner than later" and reconfirmed his commitment to take it publicly.
Both parties in Georgia face turnout challenges in the midst of the pandemic and without the polarising Trump at the top of the ballot to turn out voters from his deeply loyal supporters, as well as from detractors with deep animosity toward him.
