U.S. President Donald Trump said the search for a new FBI director is moving "very rapidly" following his decision last week to fire James Comey, who was leading the investigation into Russia's meddling in last year's presidential election and links between Trump campaign aides and Moscow interests.

Trump did not elaborate when asked about the matter on Monday.

Some opposition Democratic senators are suggesting they will not confirm Comey's successor until a special prosecutor is appointed to carry out an independent investigation of Russian interference in the election that was aimed at helping Trump defeat his Democratic opponent, former U.S. Secretary of State Hillary Clinton.

But White House spokesman Sean Spicer said "there's frankly no need for a special prosecutor" since the Federal Bureau of Investigation that Comey headed and committees in both the Senate and House of Representatives are already conducting probes on Russia's involvement.

The Trump administration has interviewed at least eight candidates for the top FBI job, and Trump has said a decision could come before he leaves Friday on his first overseas trip as president.

Meanwhile, the president continued to make headlines and rankle lawmakers from both parties. Even some supporters criticized Trump after the White House offered changing explanations for Comey's firing and an ominous tweet by the president that said the ousted FBI leader had better hope there are no "tapes" of their conversations.

Are there tapes?

On Sunday, lawmakers urged Trump to turn over any tapes of conversations with Comey. Republican Senator Lindsey Graham said the White House must "clear the air" about whether there are any taped conversations.

"You can't be cute about tapes. If there are any tapes of this conversation, they need to be turned over," Graham told NBC's Meet the Press program.

Spicer, at Monday's briefing with reporters, deflected all questions about whether Trump has installed a taping system at the White House, neither confirming nor denying its existence.

"I do not have the foggiest idea whether there are tapes or not," Democrat Mark Warner, a member of the Senate Intelligence Committee, said Sunday. "But the fact that the president made allusions to that, and then the White House would not confirm or deny?"

"First of all," he added, "we [have] got to make sure that these tapes, if they exist, don't mysteriously disappear. So I have asked, others have asked, for the tapes to be preserved, if they exist."

Senate Intelligence Committee Vice Chairman Sen. M
Senate Intelligence Committee Vice Chairman Sen. Mark Warner (R), accompanied by committee Chairman Sen. Richard Burr speak to the media on Capitol Hill in Washington, May 11, 2017.

Democrats also are accusing Trump of attempting to thwart the FBI's investigation of alleged Russian meddling in the U.S. election, and possible ties between Moscow and the Trump campaign. Some have called for some type of independent inquiry.

Also on NBC television, Trump said Comey asked to meet with him in January so as to convey a desire to remain on the job at the FBI. But the president's account does not match the recollection of someone who interacted with Comey shortly before the FBI director dined with Trump at the White House.

Former Director of National Intelligence James Cla
Former Director of National Intelligence James Clapper testifies about potential Russian interference in the presidential election before the Senate Judiciary Committee on Capitol Hill, Washington, D.C., U.S., May 8, 2017.

James Clapper, former national intelligence director who served under the Obama administration from 2010 until Trump's inauguration, told ABC's This Week, "I spoke briefly with Director Comey about the dinner. He conveyed to me that he had been invited and he was - this is my characterization — uneasy with it, simply because of the optic or the appearance of potentially compromising his independence and that of the Bureau [FBI]."

He also had harsh worlds for the president's actions toward Comey.

"I think in many ways our institutions are under assault," Clapper told CNN's State of the Union Sunday. "Both externally — and that's the big news here — is Russian interference in our election system. And I think as well our institutions are under assault internally."

When asked to clarify if the internal assault came from the president directly, the former spy chief added: "Exactly."

Preet Bharara, former U.S. attorney for the Southern District of New York who was fired by Trump, said Sunday in a Washington Post opinion piece that in order to restore faith in the system, there needs to be a bipartisan investigation in Congress as well as a special prosecutor to probe Russia's election activities.

"Congress is a check and a balance," he wrote, "and never more important than when a bullying chief executive used to his own way seems not to remember the co-equal status of the other two branches."

He also praised Comey as someone who is willing to "say no to the president" and called for the choice of the next FBI chief to be someone "apolitical" and without a partisan resume.

An NBC/Wall Street Journal poll released Sunday found only 29 percent of Americans approved of Comey's firing, while 78 percent support an independent commission or special prosecutor to investigate Russia's interference in the election.