Mr Comey reportedly asked the US justice department to publicly reject Saturday’s allegation, according to the New York Times and NBC.
He is said to have asked for the correction because it falsely insinuates that the FBI broke the law.
The department has not commented.
US media quoted officials as saying that Mr Comey believed there was no evidence to support Mr Trump’s allegation.
From an FBI director this is a startling rebuke of a sitting president and he will be under pressure from Democrats to voice it publicly, the BBC’s Nick Bryant reports from Washington.
The Republican president, who faces intense scrutiny over alleged Russian interference in support of his presidential bid, has offered no evidence to support his allegation that phones at Trump Tower were tapped last year.
The New York Times and NBC reports came as the White House called on Congress to investigate whether the Obama administration had abused its powers.
Both Congress and the FBI are currently investigating contacts between the Trump election campaign and Russian officials, after US intelligence agencies assessed that Russia had interfered with the election to help Mr Trump win against his Democratic opponent, Hillary Clinton.
White House press secretary Sean Spicer said there had been “very troubling” reports “concerning potentially politically motivated investigations immediately ahead of the 2016 election”.
Representative Devin Nunes, the Republican chairman of the House Intelligence Committee, said in a statement that his committee would “make inquiries into whether the government was conducting surveillance activities on any political party’s campaign officials or surrogates”.
Senate Intelligence Committee Chairman Richard Burr, also a Republican, said in a statement that it would “follow the evidence where it leads, and we will continue to be guided by the intelligence and facts as we compile our findings”.
James Comey is no stranger to political controversy or dramatic interventions, our correspondent says.
In the final weeks of the presidential campaign he publicly revealed the FBI was reviewing new evidence in its inquiry into Hillary Clinton’s private email server, an announcement she believes contributed to her defeat.
Mr Trump, who spent the weekend at his Florida resort, had called the alleged tapping “Nixon/Watergate”, referring to the notorious political scandal of 1972, which led to the downfall of President Richard Nixon.
He asked on Twitter whether it was legal for a “sitting president to be wire-tapping” and referred to the allegation as “a new low”.
Earlier Ben Rhodes, who was Mr Obama’s foreign policy adviser and speechwriter, wrote in a tweet: “No President can order a wiretap. Those restrictions were put in place to protect citizens from people like you.”
A spokesman for Mr Obama said Mr Trump’s allegation charge was “simply false”.
The director of national intelligence at the time of the US election, James Clapper, has also denied there was any wire-tap on Mr Trump or his election campaign team.
Mr Clapper told NBC that he knew of no court order to allow monitoring of Trump Tower in New York.
“There was no such wire-tap activity mounted against the president-elect at the time, as a candidate, or against his campaign,” Mr Clapper said.