John Prescott says he has "no doubt" Labour will lose the next general election if MPs continue sniping over the party's leadership succession.
Mr Prescott said he had been mediating between Mr Blair and Gordon Brown through "that terrible week last week".
The deputy prime minister urged all sides to consign to "the dustbin of last week" the row over Tony Blair's decision to stand down within a year.
Members would be able to speak out during the leadership contest, he said.
Mr Prescott's comments came as efforts were stepped up to end the dispute over the timing of Mr Blair's departure from Number 10.
On Sunday Chancellor Gordon Brown declared his support for the prime minister, and Mr Blair said he accepted that Mr Brown had not been plotting to bring about his early departure from Downing Street.
Asked if the backbiting could mean Labour losing the next election, Mr Prescott told BBC Radio 4's Today programme: "I have no doubt about it if we were doing that for a few more weeks - I have no doubt it will affect us.
"We have not got the luxury of... making press comments in the way that's been last week without the electorate saying... 'we can't support this government'."
The deputy prime minister, who is in Helsinki discussing global warming with representatives from Asian and European countries, would not comment on ex-home secretary Charles Clarke's criticism of Mr Brown.
Mr Clarke accused the chancellor was of being a "control freak", lacking the courage to take tough decisions.
He also claimed Mr Brown was "deluded" about his relationship with the prime minister.
Mr Prescott said: "I agree there's been some strong comments made by Members of Parliament, but I say to my colleagues, you're one third of the election process.
"I am receiving protests constantly from party members - and so is the party itself - about saying, can we stop all this backbiting?
"After all, there is an election going to take place - you'll be able to make your decision.
"And I say, look, let's get back to the unity of this party, be proud of our record and put it across to the people."
Asked about his own future, Mr Prescott said he would also be discussing his position as Labour's deputy leader with the party's national executive committee.
Middle East mission
"I am different from Tony who is prime minister. I'm a deputy leader really. I have the title of deputy prime minister at the kind of patronage of the prime minister," he said.
"The prime minister makes these kind of decisions.
"I am talking to the NEC because they are the people who are responsible for the election of party people."
Mr Prescott was speaking as Mr Blair flew into Beirut on Monday for talks with Lebanese Prime Minister Fouad Siniora, during his tour of the Middle East.
At the same time, union leader Brendan Barber used the opening of the TUC conference to urge the government to end tussling over the Labour leadership and focus on getting social justice.
He said he believed uncertainty over the leadership had damaged the Labour Party and the government.
Some union leaders want Mr Blair to quit immediately, but Tony Woodley, general secretary of the Transport and General Workers' Union, said the process should begin next March.