The allegation - made without showing any evidence - included calling Mr Obama a "bad, or sick, guy".
"Terrible!" the President tweeted early on Saturday morning. "Just found out that Obama had my 'wires tapped' in Trump Tower just before the victory. Nothing found. This is McCarthyism!"
Soon after he added: "Is it legal for a sitting President to be "wire tapping" a race for president prior to an election? Turned down by court earlier. A new low!
"I'd bet a good lawyer could make a great case out of the fact that President Obama was tapping my phones in October, just prior to election!"
Mr Trump also suggested that Mr Obama was "sick", writing: "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!"
Despite the severity of the claim, Mr Trump did not immediately provide evidence that Mr Obama was responsible for surveillance on his property.
The President was tweeting from his Florida seaside resort, Mar-a-Lago, where he was scheduled to meet with Attorney General Jeff Sessions and Homeland Security Secretary John Kelly before a dinner with officials also including adviser Steve Bannon and White House Counsel Don McGahn.
Several conservative news outlets and commentators have made allegations in recent days about Mr Trump being wiretapped during the campaign, without offering any evidence.
Mr Obama's spokesman denied the allegations, saying any suggestion he or his staff had “ordered surveillance on any US citizen” was false.
Kevin Lewis added that a “cardinal rule” of the Obama administration was that no White House official ever interfered in any Justice Department investigations, which are supposed to be conducted free of political influence.
“As part of that practice, neither Obama nor White House official ever ordered surveillance on any US citizen. Any suggestion otherwise is false,” he said.
The statement did not address the possibility that a wiretap of the Trump campaign could have been ordered by Justice Department officials.
Under US law, a federal court would have to have found probable cause that the target of the surveillance is an “agent of a foreign power” in order to approve a warrant authorising electronic surveillance of Trump Tower.
Members of Congress said the accusations require investigation or explanation, with Democratic commentators quick to defend the former President.
Ben Rhodes, a foreign policy adviser to Mr Obama, tweeted: "No President can order a wiretap. Those restrictions were put in place to protect citizens from people like you."
Representative Eric Swalwell, who sits on the House Intelligence Committee, told Fox News: "I think this is just the president up early doing his routine tweeting.
"Presidents don't wiretap anyone. These are pursued by the Department of Justice in accordance with the FBI and signed off by a judge."
Congressman Ted Lieu added: "Mr President: If there was a wiretap at Trump Tower, that means a fed judge found probable cause of crime which means you are in deep s***."
Senator Ben Sasse, a Republican, described the allegations as serious and said the public deserved more information.
He said it was possible that Mr Trump had been illegally tapped, but if so, the President should explain what sort of tap it was and how he knew about it.
Representative Adam Schiff, the top Democrat on the House Intelligence Committee, called Mr Trump's assertion a “spectacularly reckless allegation”.
“If there is something bad or sick going on, it is the willingness of the nation's chief executive to make the most outlandish and destructive claims without providing a scintilla of evidence to support them,” he said.
It was the latest twist in a controversy over ties between Mr Trump associates and Russia that has dogged the early days of his presidency.
Mr Trump and the Kremlin refuted report by US intelligence agencies that concluded Russia hacked and leaked Democratic emails during the election campaign as part of an effort to damage Hillary Clinton.
The former national security adviser, Michael Flynn, resigned in February after revelations that he had discussed sanctions with the Russian ambassador to the US during the transition period.
Mr Flynn had promised Vice President Mike Pence he had not discussed the matter, but transcripts of intercepted communications showed that the subject had come up.
The President has a long history of making unfounded and sometimes bizarre claims and it is not the first time he has used them to attack his predecessor.
Mr Trump has been a strong proponent of the 'birther' conspiracy theory, which held that Mr Obama was not born in the US.
Using anonymous sources - which he has criticised the press for utilising - Mr Trump has at least five times alleged that Mr Obama was not born in the US and that his birth certificate was a "fraud".
The tirade came at end of a week in which Mr Trump had been praised widely for acting in a more Presidential fashion, during his first address to Congress.