Senate Judiciary Committee Chairman Arlen Specter, R-Pa., told CNN on Sunday that he thinks the best course of action would be to have Bolton's nomination come to the full Senate.
He said the matter shouldn't be decided by the Foreign Relations Committee alone because, "the Constitution says that advice and consent are the province of the Senate itself."
One committee member, Sen. Christopher Dodd, D-Conn., was asked if he thought Bolton should withdraw from consideration. "I would hope he would," Dodd told CBS' "Face the Nation."
"I think he's going to embarrass the president. I think he's going to ... have a very difficult job serving if he's confirmed narrowly by the Senate. He should withdraw or the president ought to withdraw this nomination. There are plenty of other good people who embrace his ideological views who can go up" and achieve the major changes that Bush is pressing for at the United Nations. "John Bolton is not that individual."
A White House spokeswoman, Christie Parell, said the president is sticking by Bolton.
"The president believes he's exactly the man needed at the United Nations," she said Sunday in response to Dodd's remark.
Bolton has come under fire over allegations he dressed down subordinates or behaved, as one former colleague claimed, "like a madman," when he was crossed.
In addition, there is unease over Bolton's past hostility toward the United Nations and accusations that he tried to pressure career intelligence analysts into twisting facts for political reasons.
Sen. Dick Durbin of Illinois, the second-ranking Democrat, told "Fox News Sunday" that he thought Bolton's nomination was "in trouble."
Defenders said Bolton, now the undersecretary of state for arms control and international security, has a tough-talking style that will help achieve needed reforms at the United Nations.
"Look, we're not nominating somebody here for Miss Congeniality," said Sen. Mitch McConnell, R-Ky. "We know that Bolton is a person, a forceful person, a person of strong opinion."
But Sen. Joseph Biden, the top Democrat on the committee, said he was concerned about allegations Bolton tried to intimidate intelligence analysts, and in some cases, sought to have them fired.
"The only relevance of temperament here ... that goes beyond these other questions is not whether he's a nice guy or not," Biden, D-Del., told ABC's "This Week."