Trump took to Twitter on Saturday to make the allegations.
"How low has President Obama gone to tapp [sic] my phones during the very sacred election process. This is Nixon/Watergate. Bad (or sick) guy!," Trump tweeted to his 25.9 million followers.
In another post, Trump said the alleged tapping took place in October at his Trump Tower skyscraper in New York, but there was "nothing found".
Obama's spokesman Kevin Lewis denied the claims, saying: "Neither President Obama nor any White House official ever ordered surveillance on any US citizen. Any suggestion otherwise is simply false."
Trump's administration has come under pressure from FBI and congressional investigations into contacts between some members of his campaign team and Russian officials during his bid to become the US' 45th The US.
Trump's tweets came a day after he fiercely criticised Democrats over their demands for Jeff Sessions, attorney general, to resign for failing to disclose meetings with Russia's ambassador to the US, Sergey Kislyak.
Trump called allegations that Sessions lied under oath during a confirmation hearing a "total witch-hunt".
Michael Flynn, Trump's national security adviser, resigned in February after it became apparent that he had discussed US sanctions on Russia with Kislyak before Trump took office.
Flynn had told Vice President Mike Pence he had not discussed US sanctions with the Russians, but transcripts of intercepted communications showed that the subject had come up in conversation.
Trump has often used his Twitter account to attack rivals.
For years, he led a campaign alleging that Obama was not born in the US, but later retracted that claim.
Despite the hostility, though, Trump has offered some kind words about his predecessor as he settled into the White House.
In November, after winning the election, Trump said Obama was "a very good man".
Source: News agencies