The former US leader was said to be "livid" after his successor accused him of being a "bad, or sick, guy", while failing to provide any evidence his Democratic predecessor had ordered a surveillance operation.
While Mr Obama has kept a relatively low profile since leaving office, aides told The Wall Street Journal that Mr Trump's accusation - which he made without producing any evidence to back up his claim - has crossed a boundary because it questioned the former leader's integrity and that of the office of president itself.
It follows a string of tweets from Mr Trump over the weekend, that accused the then-President of wiretapping his New York office in 2016.
"Terrible!" he said. "Just found out that Obama had my 'wires tapped' in Trump Tower just before the victory. Nothing found. This is McCarthyism!"
Terrible! Just found out that Obama had my "wires tapped" in Trump Tower just before the victory. Nothing found. This is McCarthyism!— Donald J. Trump (@realDonaldTrump) March 4, 2017
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!"
Mr Obama's spokesman, Kevin Lewis, flatly denied the suggestion that anything underhand had taken place.
"A cardinal rule of the Obama administration was that no White House official ever interfered with any independent investigation led by the Department of Justice," he said.
A cardinal rule of the Obama Administration was that no WH official ever interfered with any independent investigation led by the DOJ. pic.twitter.com/c5QD50nXac— Kevin Lewis (@KLewis44) March 4, 2017
"As part of that practice, neither President Obama nor any White House official ever ordered surveillance on any US citizen. Any suggestion otherwise is simply false."
Mr Trump flung another accusation at Obama earlier this week, tweeting: "122 vicious prisoners, released by the Obama Administration from [Guantanamo Bay], have returned to the battlefield. Just another terrible decision!"
122 vicious prisoners, released by the Obama Administration from Gitmo, have returned to the battlefield. Just another terrible decision!— Donald J. Trump (@realDonaldTrump) March 7, 2017
That claim has been proven false. 113 of the so-called 'vicious prisoners' released from extrajudicial detention were set free by President George Bush, and only nine (or six per cent) by the Obama administration.
Mr Trump's camp was quick to fire back at claims these tweets were unbecoming. "Trump's people think Obama is at war with them," the media baron and Trump ally Christopher Ruddy told The WSJ.
"This president has been under siege since Day One from both the press and Obama loyalists and he’s reacting to it. I don’t think there’s any doubt that Obama loyalists inside the administration and outside are giving Donald Trump a lot of grief and a lot of problems."