Mr Balls said he and Mr Clarke were disappointed their football team lost
Ex-Cabinet heavyweight Charles Clarke has launched a bitter attack on Gordon Brown's closest ally Ed Balls, accusing him of carrying out secret briefings.
Mr Balls, the schools secretary, had described recent criticism of the prime minister as "indulgent nonsense" by those with scores to settle.
But Mr Clarke claimed Mr Balls was "well acquainted" with the kind of anonymous briefing he was criticising.
Mr Balls made light of Mr Clarke's barbs and repeated his call for unity.
The exchanges came amid anger among Labour MPs over the axing of the 10p tax rate.
As many as 70 backbenchers and six ministerial aides have expressed concern over the abolition of the lowest income tax band, amid warnings that millions of society's poorest will be hit.
Mr Balls called for an end to the public row in an interview in The Times on Friday, to prevent the party being damaged at 1 May's local elections.
"The efforts of local councillors and shadow leaders should not be undermined by this kind of indulgent nonsense," he said.
But those comments prompted Mr Clarke, who was sacked from Tony Blair's government in May 2006, to hit back: "His injunctions about the 'indulgent nonsense' of 'private briefings against the Labour leader' certainly come from one who is well acquainted with this kind of activity.
"Such things do discredit politics and take us back to the days of faction and party-within-a-party that were so damaging in the 1980s."
In a letter to The Times, the former home secretary said: "It's certainly true that many Labour MPs, including myself are disappointed by policy decisions such as the abolition of the 10p tax rate".
They were also disappointed at "the over-bureaucratic and insensitive nature of the post office closure programme, and the problems arising from lack of preparation for a Northern Rock-style economic challenge.
"These all stem from Treasury positions with which he is very familiar," he said, referring to Mr Balls, who was Mr Brown's chief economic adviser when he was chancellor.
And in a further swipe, Mr Clarke, a former education secretary, added that he and "many" others were also disappointed with Mr Balls' education policies.
"He should stop attacking others anonymously or in code and look to his own performance and record."
Before Mr Brown became prime minister last year, Mr Clarke was a vocal critic of his leadership style and was sometimes touted as a possible Blairite challenger for the top job.
But in his Times letter, Mr Clarke described suggestions that he was considering running as a "stalking horse" against Mr Brown - to destabilise the Labour leader and flush out more heavyweight contenders - as "totally false".
During an election campaign visit in south London, Mr Balls sought to smooth over any tensions between him and Mr Clarke.
"I think Charles and I were both in a bad mood this weekend after our team, Norwich City, lost 2-1 to West Brom - and he chose to kick me rather than the dog," he said.
"But seriously the important thing in the run up to the lections is that we stay united, show that we are on the side of hard-working families and continue to deliver them, like all the new children's centres we're opening here in London and across the country."
Foreign Secretary David Miliband, who was also touted as a possible Blairite challenger ahead of last year's coronation of Mr Brown as Labour leader, issued a plea for unity in a Sunday newspaper article, warning that electoral defeat was certain if the party "argued among ourselves".