White House officials say Comey's testimony about the scope of the FBI investigation upset Trump, who has dismissed the FBI and congressional investigations as a "witch hunt". The President has repeatedly said there was no collusion.
Current and former senior intelligence officials viewed Trump's requests as an attempt by the President to tarnish the credibility of the agency leading the Russia investigation.
A senior intelligence official said that Trump's goal was to "muddy the waters" about the scope of the FBI probe at a time when Democrats were ramping up their calls for the Justice Department to appoint a special counsel, a step announced last week.
Senior intelligence officials also saw the March requests as a threat to the independence of US spy agencies, which are supposed to remain insulated from partisan issues.
"The problem wasn't so much asking them to issue statements, it was asking them to issue false statements about an ongoing investigation," a former senior intelligence official said of the request to Coats.
The NSA and Brian Hale, a spokesman for Coats, declined to comment, citing the ongoing investigation.
"The White House does not confirm or deny unsubstantiated claims based on illegal leaks from anonymous individuals," a White House spokesperson said. "The President will continue to focus on his agenda that he was elected to pursue by the American people."
In addition to the requests to Coats and Rogers, senior White House officials sounded out top intelligence officials about the possibility of intervening directly with Comey to encourage the FBI to drop its probe of Michael Flynn, Trump's former National Security Adviser, according to people familiar with the matter. The officials said the White House appeared uncertain about its power to influence the FBI.
"Can we ask him to shut down the investigation? Are you able to assist in this matter?" one official said of the line of questioning from the White House.
The new revelations add to a growing body of evidence that Trump sought to co-opt and then undermine Comey before he fired him May 9.
According to notes kept by Comey, Trump first asked for his loyalty at a dinner in January and then, at a meeting the next month, asked him to drop the probe into Flynn. Trump disputes those accounts.
Current and former officials said either Trump lacks an understanding of the FBI's role as an independent law enforcement agency or does not care about maintaining such boundaries.