"It is critical to our shared efforts to counter terrorism".
In Fleming's in-tray will be to fix ties strained by claims made by the presidential press secretary, Sean Spicer, that GCHQ had helped the previous president spy on Donald Trump.
A former British ambassador to the U.S. has lashed out at Donald Trump, calling his allegations that the United Kingdom wiretapped the Trump Tower "absurd" and "nonsensical", while warning that the controversy could damage the two countries' close relationship.
Sir Peter Westmacott, who was the UK's top diplomat in the USA until January, also urged the White House to pubicly accept British assurances that the allegation about GCHQ is nonsense.
He added that "gratuitously damaging" the nations' alliance with lies without setting the "record straight would be a gift to our enemies they could only dream of".
Westmacott also referred to Trump's "famous reluctance to admit mistakes".
The unusual move by the agency came after White House Press Secretary Sean Spicer cited claims first made on US TV channel Fox News earlier this week.
The claim brought a rare public denial from GCHQ which described the suggestion as "utterly ridiculous".
Downing Street said it had secured an assurance that the allegation would not be repeated.
But he has provided no evidence for the claim. "I didn't make an opinion on it", Trump said.
Fox News has disassociated itself from the allegations, saying there is no evidence to back them. "So you shouldn't be talking to me, you should be talking to Fox".
According to documents leaked by Edward Snowden in 2013, which were reported on by the Guardian and other media outlets, Merkel was among the world leaders the NSA targeted for surveillance.
Trump also hinted at the unsubstantiated allegation during a press conference with Germany's Angela Merkel, suggesting both of them had been surveiled by USA intelligence services - a reference to reports that United States spy agency the NSA had tapped Merkel's phone.
The committee is holding a hearing on the issue Monday.
Writing in The Guardian, Sir Peter, who only retired past year, urged Foreign Secretary Boris Johnson to take up the issue when he visits Washington later this week.