Defeated Green Party candidate Jill Stein is asking for recounts in Wisconsin, Michigan and Pennsylvania where Mr Trump won narrow victories.
The next US President wrote on Twitter that the "badly defeated and demoralized" Democrats were now joining the Green Party campaign for recounts.
He said: "The Democrats, when they incorrectly thought they were going to win, asked that the election night tabulation be accepted. Not so anymore!"
Will the recounts change anything?
If Mr Trump's victories in all three states are overturned then the election would tip in Mrs Clinton's favour.
This is because the three states offered a total of 46 votes in the all-important Electoral College.
Overall Mr Trump won 370 Electoral College votes, well above the winning threshold of 270 and Mrs Clinton's 232.
But it is extremely unlikely that the recounts will uncover enough evidence of electoral fraud to overturn the results.
For Ms Clinton to win all three states, it thought that she would require a swing of more than 100,000 votes across the three states.
Mr Trump won Wisconsin by just over 22,500 votes, Michigan by about 10,700 votes and Pennsylvania by slightly more than 68,000.
Even Ms Stein and Mrs Clinton's campaign have admitted that the recounts are unlikely to chance the outcome of the US election 2016.
The Green Party candidate said: "These recounts are part of an election integrity movement to attempt to shine a light on just how untrustworthy the US election system is."
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