Trump told reporters aboard the Air Force One as he embarked on a visit to Lynchburg, Virginia that a decision may come as soon as before he starts his first oversea trip next Friday.
The job requires Senate confirmation.
Mr Trump has said he removed Mr Comey because he was not doing a good job and that Mr Comey had lost the support of Federal Bureau of Investigation employees - something rejected by the acting head of the agency.
When asked, "Internally, from the president?" So far 14 people - lawmakers, attorneys and law enforcement officials among them - have emerged as candidates.
It occurs sometimes in business meetings or phone calls, and sometimes recordings resurface as transcripts in court, said Stephen Meister, a NY real estate attorney who has represented Mr. Trump in the past and said he had no knowledge of Mr. Trump recording phone calls.
Media captionWhat do Trump supporters think about Comey's firing?
"You can't be cute about tapes", the SC lawmaker told NBC.
CNN said Comey is "not anxious about any tapes" Trump may have, citing an unnamed source familiar with the matter.
Among other roles, Townsend led the Office of Intelligence Policy and Review, which at the time handled requests for government surveillance warrants in terrorism and espionage cases.
Mr Trump's press secretary Sean Spicer has rejected the report. Mark Warner, D-Va., top Democrat on the Senate Intelligence Committee, called the insinuation "outrageous" during his appearance on "Fox News Sunday".
Earlier this month, he questioned on Twitter when the "Fake Media" would ask about the Democrats' dealings with Russian Federation and why the fake news declines to report that ObamaCare is "on its last legs". "I have asked, others have asked, to make sure the tapes are preserved, if they exist".
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"I actually said today, let's not ever do any more press briefings", Trump told Fox News.
Why did Trump fire Comey?
This could potentially trigger a showdown with the White House as Democrats consider a boycott of the vote on Comey's replacement.
Trump wrote to Comey that the dismissal was necessary to restore the public's trust and confidence. "I would find that very inconsistent with what I know of Jim Comey", Clapper told MSNBC, adding it would have been "out of character for him to ask to stay on". So it's possible Spicer just plain doesn't know whether Trump is taping conversations - and has no interest in finding out.
The White House had no immediate comment and no White House aide appeared on the Sunday news shows, leaving Nikki Haley, the US ambassador to the United Nations, to defend Trump.
"As a very active President with lots of things happening, it is not possible for my surrogates to stand at podium with flawless accuracy!"
His suggestion to tackle the problem, he said, was: "We don't have press conferences". Eight met May 13 with Attorney General Jeff Sessions and his deputy, Rod Rosenstein, throughout the day. In the NBC interview on Thursday, Trump said he would have fired Comey regardless of any such recommendations.
Warner said he would "absolutely" want the Senate to subpoena the White House to produce any recordings made by the Trump administration that are relevant to the Comey firing or the Russian Federation matter.
President Donald Trump is considering at least a dozen people to succeed ousted FBI Director James Comey.
As for the FBI investigation into possible Trump team collusion with the Russians, the president, after calling it "a total hoax" as well as a "taxpayer charade", told Mr. Holt he wanted it to be "absolutely done properly" and to "get to the bottom" of the alleged conspiracy. Moscow has denied any such meddling.
Pratt Hotel Corp., which owned the Sands casino in Atlantic City, filed a lawsuit against Mr. Trump's casino company in 1989, claiming it had interfered with Pratt's attempts to buy a construction site next to the Trump Plaza casino.