Chaffetz said Tuesday that he doesn't want to say whether, if true, such a request by Trump would be obstruction of justice, but he wants to see the memo and hear from Comey.
"If that memo exists, I want to see it sooner rather than later," Chaffetz said in an interview. "It causes great concern if the reporting is true."
"Given the recent actions, it's ripe for Oversight [Committee]," Chaffetz added.
Chaffetz asked for the memo Tuesday as part of a broader request to James McCabe, the acting FBI director, for all documents related to Comey's conversations with the president. He set a deadline of May 24.
Trump fired Comey on May 9, later saying that he wanted the FBI investigation to end.
Flynn, a retired Army general, was fired after misleading the White House about his discussions with Russian Ambassador Sergey Kislyak.
Chaffetz said if the FBI didn't provide the requested information he would issue a subpoena; the Oversight Committee chairman has unilateral subpoena power.
Chaffetz said he wasn't sure what his probe may find but that he found the concept of the president asking the FBI to stop an investigation troubling.
"There are a myriad of possibilities, and I think it's irresponsible to jump to any one conclusion," Chaffetz said. "It is cause for concern. We do have a constitutional duty to provide such oversight, and I plan to help do that."
Sen. Orrin Hatch, R-Utah, said Tuesday night that he hadn't seen the Times story, but he noted that Trump was at one point irritated with Flynn for misleading the White House and probably wouldn't be doing him any favors.
"I doubt that he asked him to do that," Hatch said of Trump asking Comey to halt any investigation.
Even after firing Flynn, Trump had praised him as a "good man."
Tim Chambless, a political science professor at the University of Utah, said he could draw a parallel between Trump reportedly asking Comey to end an investigation and President Richard Nixon's efforts to halt an FBI probe of his activities in the Watergate scandal.
Chambless said this latest news smacks of obstruction of justice and a cover-up.
"It's Nixonian to me," Chambless said.
Meanwhile, Chaffetz said Tuesday that he wasn't likely to pursue asking the White House for any recordings it may have of Trump's conversations with Comey or others. Trump has hinted there may be such recordings when he warned on Twitter that Comey had better hope that there aren't any "tapes" of their discussions.
"I'm going to leave that door open," Chaffetz said of requesting possible recordings. "I'm not saying absolutely no, but we want to find out about what memo or other documents exist at the Department of Justice. That's where we're going to be looking."