The US president suggested he was the victim of the same sort of surveillance the Obama administration was once alleged to have used to monitor German Chancellor Angela Merkel's calls.
Donald Trump also stood by his unproven claim that his predecessor Barack Obama wiretapped his phones.
Karl repeatedly pressed Spicer over the White House's failure to produce any evidence supporting the major claim.
"There's widespread reporting that throughout the 2016 election, there was surveillance that was done on a variety of people", Mr. Spicer said.
The Government Communications Headquarters (GCHQ) responded to White House claims, which were repeated by Press Secretary Sean Spicer on Thursday, after being first mentioned on Fox TV earlier on Tuesday.
He said: "We have made clear to the (US) administration that these claims are ridiculous and that they should be ignored and we have received assurances that these allegations won't be repeated".
"That was a statement made by a very talented lawyer on Fox".
"You shouldn't be talking to me".
"This shows the administration doesn't give the allegations any credence", the Downing Street spokesman said.
Tim Farron, leader of the UK Liberal Democrat party - the junior coalition partner in the last British government - described the White House claim as "shameful" and said it risked harming U.S. and UK security. Charlie Dent called the accusation against Britain "inexplicable" and the accusation against Mr Obama unfounded.
The rebuke comes a day after the House Intelligence Committee offered a similar assessment, leaving the White House virtually alone in asserting the surveillance claim. He did not elaborate on the details of the information.
Chairman Devin Nunes (R-Calif.) and ranking member Adam Schiff (D-Calif.) originally requested the Justice Department send them copies of any wiretapping warrants, applications, court orders or other proof of Trump's allegations by this past Monday.
Now Grassley is threatening to hold up the nomination of Rod J. Rosenstein to be deputy attorney general until the committee is fully briefed by the FBI about whether it has begun a full criminal investigation into Russian interference in last year's elections. "I just don't want anybody jumping to any conclusion on any side of this, other than the fact that we know that Obama didn't physically wiretap Trump Tower", Nunes told CNN Friday. "Nothing found. This is McCarthyism!" he tweeted.
Is it legal for a sitting President to be "wire tapping" a race for president prior to an election? "A NEW LOW!" he continued, accusing Obama of crimes comparable to those of Richard Nixon's Watergate scandal.