They believe that only the sacrifice of one or more of the big beasts of the jungle, such as Cheney or Donald Rumsfeld, the defence secretary, will convince voters that Bush understands the need for a fresh start.

The jittery Republicans claim Bush’s mini-White House reshuffle last week will do nothing to forestall the threat of losing control of Congress in the November mid-term elections.

Fred Barnes of the Weekly Standard magazine and author of Rebel in Chief, a sympathetic new biography of Bush, said: “There are going to have to be sweeping personnel changes if people are going to take a second look at the Bush presidency.”

Barnes, who is close to the White House, said he believed Cheney would be willing to stand down in order to help Bush. “It’s unlike Bush to dump somebody whom he likes and respects,” he cautioned. “But the president needs to do something shocking and dramatic such as putting in Condoleezza Rice.”

Cheney appeared to have beeen caught napping during a visit to the Oval Office by China’s president, Hu Jintao, on Friday, although he claimed he had been looking down at his notes. It has often been said that he would cite medical reasons should he ever resign.

The best scenario, Barnes added, would be for Bush to announce that “Dick Cheney will be around as an outside adviser and I can call him on the phone, but I’d like to anoint somebody who I think will be the next leader of the United States”.

Tom Edmonds, a leading Republican consultant, said the White House had failed to grasp that the party was in desperate straits. “I have never talked to so many disenchanted Republicans,” he said. “The president even stonewalled the minor changes he made by talking about how he was really perfectly happy with his team. He didn’t even give himself wiggle room.”

One Republican strategist, who did not want to be named, said: “If I were Bush I would think of changing Cheney. It is one of the few substantial things he can do to change the complexion of his administration. The rest is nibbling around the edges.”

Bush’s new chief of staff, Joshua Bolten, quickly put his stamp on the inner workings of the White House last week by stripping Karl Rove, Bush’s most powerful adviser, of his policy-making role and ordering him to concentrate on his forte: winning elections.

Bolten also obtained the resignation of the White House spokesman, Scott McClellan, who was nicknamed Piggy in a recent Vanity Fair article because of his resemblance to the hapless victim of the feral boys in Lord of the Flies.

Tony Snow, a Fox News broadcaster who is favoured to replace McClellan, has previously described the Bush administration as “listless” and in dire need of change.

But a new communicator cannot reinvent an old team. Edmonds believes Rumsfeld should go. “The president is loyal to a fault,” he said. “His loyalty shouldn’t be to Rumsfeld but to the troops in Afghanistan and Iraq. We need a new, strong face on the war, such as Senator John McCain or Joe Lieberman (the pro-war Democrat senator).”

Bob Schieffer, a CBS news television presenter, said Bush may yet drop Rumsfeld despite his strong declaration of support. “It was also this president who said, ‘Brownie, you’re doing a heck of a job’ and that was just before Brownie got canned,” Schieffer said, referring to Michael Brown, who directed the Federal Emergency Management Agency’s much-criticised response to Hurricane Katrina.

John Snow, the Treasury secretary, has been left twisting in the wind while replacements for him are openly discussed, and Rob Portman has been brought in to replace Bolten as budget director. Suggestions that Harriet Miers, Bush’s White House counsel who was dropped as his supreme court nominee, would be next to go were denied last week.

Supporters say Bush should live up to his bold claim that he is “the decider” — made while rejecting recent calls for Rumsfeld’s resignation from half a dozen senior generals — and start firing senior people rather than backroom staff.

“If the Democrats win either the House of Representatives or the Senate it will be death and torment. It will be horrible for Bush,” said Barnes. A Democrat win could lead to moves to impeach Bush for leading the country to war on allegedly false pretences, or at the very least, to bog down the president’s legislative programme until he leaves office in 2008.

Rove has been privately warning party activists to expect some losses in the mid-term elections. One insider said: “I’ve heard him say at several party gatherings that the president wasn’t supposed to win in 2000, but he did. We’ve increased our margins of victory time and again. We can’t just keep winning on top of winning so we’re bound to slip back, but we’re still doing better than you would historically expect.”

Only one two-term victor has been more unpopular than Bush at a similar six-year stage in his presidency — Richard Nixon in the months before he was impeached.

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