Getting rid of Nick Clegg would "damage the Lib Dems for ever", former party leader Lord Ashdown has said.
Two Lib Dem parliamentary candidates in key swing seats have urged Mr Clegg to step down after the party lost 307 councillors in Thursday's election.
But Lord Ashdown said this would be "divisive", not "serious politics" and said Mr Clegg was "the best Prime Minister Britain hasn't got".
Tim Farron also warned Lib Dems would be "foolish" to turn on themselves.
On Saturday Jackie Porter, would-be MP for Winchester, said that after another poor local election showing, it was time to act in the party's best interests.
And West Dorset candidate Ros Kayes also said the public had lost trust in Mr Clegg. Both Winchester and West Dorset are currently held by the Conservatives but are seen as winnable by the Lib Dems.
Mr Farron, Lib Dem president and the MP for Westmorland and Lonsdale, said he understood a lot of people were "bruised" by the local election results on Thursday and Friday.
"I've lost elections before and it's miserable," he told the BBC's Andrew Marr show.
"I also understand why many people will feel that the message they got back on the doorsteps - that they didn't like us in government - is a really tough one."
He added: "But I just think that at this time it would be absolutely foolish of us as a party to turn in on ourselves."
Former leader Lord Ashdown said he respected the opinions of party members but believed calls for a change of leadership were "ridiculous".
"This idea that is being put about by these people who are calling for a leadership election is just about the silliest idea I have heard in my political career," he told the BBC's Sunday Politics.
A leadership contest a year away from the election would be "totally unnecessary and divisive", he added.
"It is not serious politics," he added. "This is the moment we need to get out with a really good message and campaign through the summer in the context of a general election."
Over the next year, he suggested, voters would be asking themselves whether the Lib Dems could "hack" being in government.
"If we were to take that step, I think the answer would be incontrovertibly no and that would damage the party for ever."
Following the local election results on Friday, all three established parties have faced questions about their reaction to the share of the vote achieved by the UK Independence Party.
Deputy Labour leader Harriet Harman said with 300 new councillors following the elections, she would be "baffled" at her party turning in on itself.
She said Labour had had a great set of elections on Thursday and rejected suggestions that Labour leader Ed Miliband was less popular than his party, or that the party should have done better.
Ms Harman said Mr Miliband was the only leader to have "cognised" people's views on things like his cost of living campaign and had changed the agenda. She told the BBC's Andrew Marr Show people should "look at what he has done".
She said predicting next year's General Election by comparing these local election results with previous years was "for the birds" given the "unprecedented situation" of the Tories and Lib Dems in coalition, and UKIP having gained ground.
"I'm sorry I'm not able to be here to tear my hair out and agree we are emerging blinking from the rubble... that might be the case for the other parties but it's not the case for us," she said.
Ms Harman added that the message from UKIP's vote showed that people had concerns, and that Labour was "determined to address them".
Meanwhile, Home Secretary Theresa May said the "only leadership question in politics today" was whether David Cameron or Ed Miliband would be elected prime minister in 2015.
The MP dismissed suggestions, put to her on the Andrew Marr Show, that she was "arm-wrestling" George Osborne and Boris Johnson to take Mr Cameron's place as party leader.
She said the party was focused on "getting on with the job" of looking ahead to the next general election.