"Somebody ought to be accountable," Richard Shelby, the ranking Republican on the powerful Senate Intelligence Committee told CNN. "If I were the president, he wouldn't be there."
In a surprise statement Friday, Tenet took responsibility for inclusion in Bush's January 28 State of the Union Address of an erroneous allegation that Iraq sought to buy nuclear materials in Africa.
Other congressional lawmakers said Sunday they will await the findings of an investigation into the matter before taking a position on Tenet's future. The Senate, by voice vote, has approved a probe into the matter.
"Well, I would like to wait for the end of the investigation to reach a conclusion as to whether Tenet should go. I'm obviously dissatisfied with him in this regard, but also in other aspects as well," Senator Carl Levin, the ranking Democrat on the Senate Armed Services Committee, told CNN.
Levin directed fire, however, at how the remarks came to be included in a presidential speech to the nation.
"That is highly misleading. It is intended to create a false impression. And someone in the White House was pushing the CIA," he said.
But interviewed on the same program, Bush's national security adviser Condoleezza Rice defended the CIA spy chief and dismissed suggestions he should resign.
"Absolutely not...George Tenet is a fine director of central intelligence. He has fought the war on terrorism well," Rice said.
Saturday Bush, who has just returned from an African tour, publicly backed the head of the Central Intelligence Agency.
"I've got confidence in George Tenet, I've got confidence in the men and women who work at the CIA. I look forward to working with them... as we win this war on terror," Bush told reporters as he met Nigerian President Olusegun Obasanjo on the final leg of a five-nation African tour.
Despite this, Democratic lawmakers made it clear they are not going to let the matter rest with Bush administration expressions of confidence in Tenet.
"If we do not know what our intelligence community is telling us and whether or not it is broadly true, we have a serious problem," Democratic senator, and 2004 presidential candidate, John Kerry told CNN.
"No, the case is not closed," Kerry added.
The 50-year-old Tenet was named head of the secretive intelligence agency in July 1997 by then president Bill Clinton and was kept in the post by Bush.