The US justice department and the CIA's internal watchdog announced a joint inquiry into the destroyed tapes on Saturday, to determine whether a full investigation is warranted.
But Joe Biden, a Democratic presidential candidate and chairman of the Senate Foreign Relations Committee, called for a special counsel to investigate the matter instead.
Speaking about Michael Mukasey, the attorney-general who heads the justice department, Biden said: "He's the same guy who couldn't decide whether or not waterboarding was torture and he's going to be doing this investigation.
"It appears as though there may be an obstruction of justice charge here, tampering with evidence, and destroying evidence," he said.
"I think this leads right into the White House. There may be a legal and rational explanation, but I don't see any on the face of it.
"I just think it's clearer and crisper and everyone will know what the truth is ... if he appoints a special counsel, steps back from it."
Chuck Hagel, a Republican senator, said the intelligence committees in the senate and the House of Representatives were investigating the destruction of the tapes to see if justice was obstructed and who in the White House might have known about the tapes' destruction.
Michael Hayden, the CIA director who is to testify before congress on Tuesday, told staff last Thursday that the recordings were destroyed out of fear the tapes would leak and reveal the identities of interrogators.
But Devon Chaffee, a human rights lawyer based in Washington, told Al Jazeera it was "difficult to swallow the explanation", given the timing of the destruction and the controversy of the harsh interrogation programme.
Dana Perino, the White House press secretary, said Bush "has no recollection" of hearing about either the tapes' existence or their destruction before being briefed about it on Thursday morning.
The White House has declined to comment on one news report that Harriet Miers, the then White House counsel, knew about the CIA's planned destruction of the videotapes in 2005 and urged the agency not to do it.