Following Rogers pointing out that it would be a violation of United States law to do so and that American intelligence has a close relationship with Britain, Rogers was then asked if he believed that this claim does damage to the relationship.
Asked if the NSA had requested GCHQ to monitor Mr Trump, Admiral Rogers said: "No sir, nor would I, that would be expressly against the construct of the Five Eyes agreement".
He warned it risked undermining a relationship that was "critical" to US-UK efforts to counter a range of threats including terrorism, Russian aggression, Chinese cyber-attacks, and the nuclear threat from North Korea.
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.
Westmacott, who was the UK's ambassador to Washington from 2012 to 2016, says "gratuitously damaging [the relationship] by peddling falsehoods and then doing nothing to set the record straight would be a gift to our enemies they could only dream of".
"So you have no intelligence that suggests or evidence that suggests any votes were changed?"
Westmacott also wrote disdainfully about Trump's "famous reluctance to admit mistakes".
Injured Pogba ruled out of Manchester United's trip to Boro
The job, reportedly, is Agnew's to lose at the Riverside with chairman Steve Gibson putting faith in him. Jesse Lingard's relief was palpable after Manchester United finally escaped sixth place.
White House press secretary Sean Spicer also accused British intelligence of tapping Trump's calls at Obama's behest.
GCHQ responded by saying the allegations were "nonsense, utterly ridiculous and should be ignored".
When questioned about the stunning accusation during a meeting with Chancellor Angela Merkel on Friday, Trump said Spicer was simply repeating a media report.
It led to USA officials making an apology for repeating the claims which had been initially made on Fox News. I didn't make an opinion on it.
Fox News has disassociated itself from the allegations, saying there is no evidence to back them.
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.