In a series of tweets, Mr Trump said he “just found out that Obama had my ‘wires tapped’ in Trump Tower just before the victory”.
In another tweet, he added: “Is it legal for a sitting President to be ‘wire tapping’ a race for president prior to an election? A NEW LOW!”
He also said: “How low has President Obama gone to tapp my phones during the very sacred election process. This is Nixon/Watergate. Bad (or sick) guy!”
He offered no evidence or said what prompted the allegation.
Mr Obama’s office is yet to respond.
It is thought Mr Trump’s tweets could be in response to Democrats’ outcry following revelations that Attorney General Jeff Sessions - in his confirmation hearings - did not disclose his contacts with Russia’s American ambassador during the campaign.
Mr Sessions, who was a senator at that time, was Mr Trump’s earliest Senate supporter.
US intelligence agencies have concluded Russia influenced the election to help Mr Trump win.