The President-elect claimed the recount effort "is a scam by the Green Party"
A representative of the Hillary Clinton campaign has said they will participate in the recount process, while Donald Trump has insisted "the election is over".
It comes after Green Party candidate Jill Stein earned just under $6 million to fund recount applications in three key swing states where Donald Trump won by narrow margins of less than 1%.
The election commission in Wisconsin has already received a request for a vote recount, with Stein pledging similar requests for Pennsylvania and Michigan.
Stein said "the unexpected results of the election and reported anomalies need to be investigated".
It came after a number of media outlets reported that a prominent computer scientists and election lawyers have been investigating the possibility that electronic voting machines could have been manipulated or hacked.
J Alex Halderman - Professor of Computer Science at the University of Michigan - wrote in a blog post: "Examining the physical evidence in these states - even if it finds nothing amiss - will help allay doubt and give voters justified confidence that the results are accurate".
In a post on Medium today, Clinton campaign counsel Marc Elias wrote: "Because we had not uncovered any actionable evidence of hacking or outside attempts to alter the voting technology, we had not planned to exercise this option ourselves, but now that a recount has been initiated in Wisconsin, we intend to participate in order to ensure the process proceeds in a manner that is fair to all sides.
"If Jill Stein follows through as she has promised and pursues recounts in Pennsylvania and Michigan, we will take the same approach in those states as well. We do so fully aware that the number of votes separating Donald Trump and Hillary Clinton in the closest of these states - Michigan - well exceeds the largest margin ever overcome in a recount."
He adds that "we feel it is important, on principle, to ensure our campaign is legally represented in any court proceedings and represented on the ground in order to monitor the recount process itself".
President-elect Donald Trump, however, has criticised the recount effort.
"The people have spoken and the election is over, and as Hillary Clinton herself said on election night, in addition to her conceding by congratulating me, 'We must accept this result and then look to the future," the Trump team said in a statement quoted by CNN.
"This recount is just a way for Jill Stein, who received less than one percent of the vote overall and wasn't even on the ballot in many states, to fill her coffers with money, most of which she will never even spend on this ridiculous recount.
"This is a scam by the Green Party for an election that has already been conceded, and the results of this election should be respected instead of being challenged and abused, which is exactly what Jill Stein is doing," the statement adds.
During the campaign, Trump himself had repeated claims that the election was likely to be 'rigged', and he refused to say whether he would accept the result in the event of a Clinton win.