Labour MP Gordon Prentice has urged Gordon Brown to resign, arguing the government has "gone a bit rusty".
He said the prime minister lacked the skill "to persuade and enthuse" and should go in Labour's "best interests".
But the PM was backed by a number of ministers as debate over his position continued in the wake of the party's Glasgow East by-election defeat.
And Parliamentary Labour Party chairman Tony Lloyd said Mr Prentice did not reflect the views of most Labour MPs.
Mr Lloyd told the BBC the "widespread view" among Labour MPs was not that Mr Brown should go - but that there was a need for a "sharper focus" and for Mr Brown "to get across what he's all about".
Labour saw its 13,507 majority in Glasgow East overturned by the Scottish National Party in last Thursday's by-election.
That result followed the recent loss of the Crewe and Nantwich seat, poor local elections and continuing low opinion poll showings.
There have been reports of unhappiness with Mr Brown, although so far only Mr Prentice and one other Labour MP, Graham Stringer, have spoken publicly of the need for a change of leader.
Mr Prentice, MP for Pendle, who was one of the few Labour MPs not to back Mr Brown to replace Tony Blair last year, said: "A prime minister needs a different set of skills from a chancellor of the exchequer.
"A prime minister must be able to communicate, persuade and enthuse. If not, the message is lost."
He added: "I hope Gordon reflects on things during August and accept that it is in the party's best interests, and perhaps his own, for him to stand down."
Mr Prentice told the BBC he wanted to see a leadership election: "For over a decade we have had no elections. It is no wonder we are a bit rusty."
The reason I so strongly support him is because the big problems people are facing in this country at this moment are the economy, the cost of fuel and food prices
She said people had not yet seen "the best of" Mr Brown as prime minister.
"But the reason I so strongly support him is because the big problems people are facing in this country at this moment are the economy, the cost of fuel and food prices.
"And I think Gordon Brown, more than anybody, has done more over the last 10 years to make people better off."
Ms Harman added that many of the problems had arisen from the international economic downturn, and said he was the "solution" rather than the "problem".
Several other leading government figures, including Justice Secretary Jack Straw, Schools Secretary Ed Balls and Cabinet Office Minister Ed Miliband, have publicly defended Mr Brown.
On Sunday, former deputy prime minister John Prescott demanded an end to "pointless" manoeuvring against Mr Brown - and insisted no other minister was "anywhere near" capable of taking over at Number 10.
But Paul Kenny, leader of the GMB union, urged Labour MPs to trigger a ballot ahead of the party's annual conference in September.
Following the by-election result, he said it was "better to have the ballot" and "see where we go".
"The MPs have got to make a strong decision as to whether they want to go into an election with Gordon Brown or have a [leadership] contest," he added.
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