Trump ignored calls by members of his own Republican Party to drop the claim and apologise, amid worries that his credibility and that of the United States are suffering as a result, and that his wider agenda could be derailed.
Speaking at a joint press conference with German Chancellor Angela Merkel, the president answered a question on the wiretap allegation by referring to the US National Security Agency's reported tapping of Merkel's phone several years ago.
"As far as wiretapping I guess by this past administration, at least we have something in common perhaps," Trump said.
The offhand comment came after the White House was forced into damage control mode, backing off an allegation that Britain's GCHQ spy agency had been recruited by Obama to monitor communications at Trump Tower in New York.
The midtown Manhattan skyscraper houses Trump's residence and offices, during last year's presidential election.
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No evidence of Trump wiretap: Senate panel
'McCarthyism!'
The controversy was created by Trump himself, in a series of tweets early on a Saturday morning, March 4. The property mogul-turned-politician was spending a weekend at his Mar-a-Lago estate in Florida after a politically bruising week.
"Terrible! Just found out that Obama had my 'wires tapped' in Trump Tower just before the victory. Nothing found. This is McCarthyism!" he wrote.
"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!" he continued, accusing Obama of crimes comparable to those of Richard Nixon's Watergate scandal.
The accusations - flatly denied by Obama - were at first seen as frivolous, spur-of-the-moment comments by a man who for years has often sent out colorful tweets targetting everything from the sex lives of actresses to why windmills are bad.
But two weeks later, the White House has doubled down. It has not retracted the allegations - the tweets remain online -- nor has it offered any evidence.
Trump not backing down
Trump told Fox News late Wednesday that he was going to produce proof - eventually.
"Let's see whether or not I prove it. I just don't choose to do it right now," he said. "I think we have some very good stuff. And we're in the process of putting it together, and I think it's going to be very demonstrative."
Yet few in Washington are expecting he can do that.
This week, the top four lawmakers in Congress dealing with intelligence issues, Republicans and Democrats, all said that they have been given no evidence to support Trump's allegations.
Then on Thursday, defending Trump over the wiretap claims, his spokesman Sean Spicer cited a Fox news report that tied the British spy agency into the story.
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After the GCHQ labelled the charge "utterly ridiculous" and British officials called their US counterparts to complain, the White House said that citing the Fox News claim was not an endorsement.
"Mr Spicer was simply pointing to public reports, not endorsing any specific story," the White House said in a statement.
But Trump kept that thread going as well Friday, praising the Fox News commentator who started it, former New Jersey judge Andrew Napolitano, as a "very talented legal mind" - while also not explicitly endorsing his claim.
In an official statement, Fox said it cannot confirm Napolitano's reporting, and "knows of no evidence of any kind" that Trump was surveilled.
Some Republicans insisted Friday that Trump back down.
"It's inexplicable," said Representative Charlie Dent. "I think he should simply retract it."
"I see no indication that that's true," Representative Tom Cole told reporters. "Frankly, unless you can produce some pretty compelling proof, then I think president Obama is owed an apology."
