Hours before a Wednesday deadline, the Trump campaign said it wanted a recount in the counties of Milwaukee and Dane, alleging irregularities.
Mr Trump has been making unsubstantiated claims of fraud and refused to authorise a handover.
Mr Biden says delaying the transition will damage the US pandemic response.
The Trump campaign has filed a flurry of lawsuits contesting the results in key states, although election officials say there is no evidence of widespread irregularities.
President-elect Joe Biden, a Democrat, is projected to have won the popular vote by more than 5.6 million ballots - 3.6 percentage points. In the US electoral college system that decides the presidency, he has 306 votes to Mr Trump's 232.
The president would have to overturn results in at least three states to win the election, which analysts say would be unprecedented. A recount is currently under way in Georgia where Mr Biden is 14,000 votes ahead.
What's happening in Wisconsin?
Under Wisconsin law, Mr Trump has the right to request a recount because the margin of Mr Biden's win was less than 1% but greater than 0.25%. However, his campaign must first cover the expenses of the operation.
State officials said on Wednesday they had received $3m (£2.2m) from the Trump campaign to cover the costs of the recount, which is expected to take about two weeks.
The Wisconsin Elections Commission said on Monday that a full recount would have cost an estimated $7.9m.
In its request for a partial recount, the Trump re-election campaign alleged that absentee ballots - those made by post - had been altered and improperly issued, and voter identity laws had been circumvented. It did not provide any evidence.
Milwaukee and Dane, both traditionally Democratic-leaning areas, together represent more than a third of Mr Biden's Wisconsin votes, the Milwaukee Journal Sentinel reported. In Milwaukee, Mr Biden won by 317,270 votes to 134,357 while in Dane County the margin was 260,185 to 78,800 in Mr Biden's favour.