The Cabinet reshuffle shows the government has lost its authority and is in "terminal decline", Conservative leader David Cameron has said.
Some MPs are said to be calling for a departure date from Mr Blair
As some Labour backbenchers urged Tony Blair to name his departure date, Mr Cameron said MPs were plotting rather than thinking about policy.
Mr Blair has held talks this weekend with his likely successor, Gordon Brown, about Labour's future.
Mr Brown has again called for the urgent renewal of the Labour Party.
In an interview to be broadcast on GMTV's Sunday programme, he said the local election results had been a wake-up call for the government to update its policies.
"It is definitely the case that after a period in government you have got to renew yourself by being able to address not just the issues of the day but the issues of the future," he said.
"There is policy renewal and there is party organisational renewal and the two things go hand in hand."
The pressure on the prime minister has grown after Labour's heavy losses in Thursday's local elections in England.
A letter urging Mr Blair to set a date for his expected handover to Mr Brown is said to be circulating among backbench MPs.
Some backbenchers have privately suggested the letter has been signed by 50 MPs.
'Attacking each other'
The prime minister has said he will stand down before the next general election but refuses to set a timetable.
He has signalled his intention to stay on by making sweeping changes to his Cabinet, including replacing his beleaguered home secretary and his foreign secretary.
John Prescott has also been stripped of his department, although he stays on as deputy prime minister and retains his salary and grace-and-favour homes.
Mr Cameron said the fact that this weekend's talks between Mr Blair and his chancellor were leading the news bulletins showed the depth of government divisions.
He claimed that voters wanted a replacement government, not a reshuffle.
"The MPs are plotting rather than thinking about policy, they're attacking each other rather than standing up for the country," said Mr Cameron.
"We've got ministers who've been kept in office even though they've failed, some who've been kept actually with all the trappings of office in the case of the deputy prime minister, but without any real ministerial power."
There has been criticism from Labour backbenchers too.
Labour MP Graham Stringer said Mr Blair should set out a clear timetable and leave in the next 12 months.
Hazel Blears says focus on personalities is "froth"
Fellow backbencher Geraldine Smith said Mr Blair would face "enormous pressure" next week.
"In order to unify us, I think Tony Blair has got to be honest and tell us when he's going to stand down," she told BBC Radio 4's Today programme.
The Morecambe Bay MP suggested the reshuffle, seen as a purge of senior ministers, had made things worse.
Labour Chairman Hazel Blears, newly promoted to the Cabinet, denied the party was in a mess.
The public was interested in "bread and butter" issues such as schools and hospitals, not "gossip" about internal Labour Party politics, she argued.
The pressure on Mr Blair comes as he celebrates his 53rd birthday and the first anniversary of his third general election victory.
Mr Brown is expected appear on the BBC's Sunday AM show as Labour debates how to fight back after a bad night at the polls.
The chancellor has already warned that the government needs to act to renew itself "within days".
He described the local elections, when Labour lost more than 300 council seats, as a "warning shot".
Two former ministers and Brown supporters, Andrew Smith and Nick Brown, have also called for a public timetable for a leadership succession.
Mr Smith told BBC News: "I think the sooner we see a timetable for the orderly transition which the prime minister has promised the better."