Gonzales has been fending off Democratic demands for his firing in the wake of disclosures surrounding the ousters of eight U.S. attorneys — dismissals Democrats have characterized as a politically motivated purge.
Support from many Republicans had been muted, but there was no outright GOP call for his dismissal until now.
"I think the president should replace him," Sununu said in an interview with The Associated Press. "I think the attorney general should be fired."
Bush, at a news conference in Mexico, told reporters when asked about the controversy: "Mistakes were made. And I'm frankly not happy about them."
Senior administration officials said that — at this point — Mr. Bush will resist calls to fire Gonzales, reports CBS News chief White House correspondent Jim Axelrod.
The president expressed confidence in Gonzales, a longtime friend, and defended the firings. "What Al did and what the Justice Department did was appropriate," he said. What was "mishandled," Mr. Bush said, was the Justice Department's release of some but not all details of how the firings were carried out.
The developments unfolded as presidential aides labored to protect White House political director Karl Rove and former counsel Harriet Miers from congressional subpoenas.
The White House dispatched presidential counsel Fred Fielding to Capitol Hill to negotiate the terms of any testimony by White House aides in an institutional tug-o-war reminiscent of the Watergate and the Iran-Contra scandals.
Sununu said the firings of the prosecutors, together with a report last Friday by the Justice Department's inspector general criticizing the administration's use of secret national security letters to obtain personal records in terrorism probes, shattered his confidence in Gonzales.
"We need to have a strong, credible attorney general that has the confidence of Congress and the American people," said Sununu, who faces a tough re-election campaign next year. "Alberto Gonzales can't fill that role."
Some of the dismissed prosecutors complained at hearings last week that lawmakers tried to influence political corruption investigations. Several also said there had been Justice Department attempts to intimidate them.
E-mails between the Justice Department and the White House, released Tuesday, contradicted the administration's earlier contention that Bush's aides had only limited involvement in the firings.
Senate Majority Leader Harry Reid, D-Nev., predicted Wednesday that Gonzales would soon be out.
"I think he is gone. I don't think he'll last long," Reid said in an interview with Nevada reporters. Asked how long, Reid responded: "Days."
White House Counsel Fielding, negotiating with lawmakers over possible administration testimony, is a veteran of the Nixon and Reagan White Houses. He was hired by Mr. Bush this year to handle just these kinds of demands by the Democratic-controlled Congress.
It was unclear whether the president would grant Democratic requests for his own aides to tell their stories under oath.
For his part, Gonzales, in a brief hallway interview with reporters, said he intended to cooperate where his aides are concerned.
"We want Congress to know, to understand what happened here. We'll go, we'll work it out," he said.
Earlier, on CBS News' The Early Show, Gonzales said, "I didn't become attorney general by quitting. Obviously, my job is easier if I have the confidence of Congress."