Liberal Democrat president Tim Farron has appealed for an end to "absolutely foolish" calls for Nick Clegg to quit as leader.
Mr Clegg's future has been questioned by MPs as Lib Dem candidates in target Westminster seats joined voices urging him to fall on his sword over dismal election results.
The Deputy Prime Minister has insisted he is staying despite the party losing more than 250 councillors in local elections and anticipating another rout when European election results emerge later.
Southport MP John Pugh suggested that a dozen of his Commons colleagues had expressed doubts over whether Mr Clegg should continue at the head of the party.
In a call for unity Mr Farron - a favourite to succeed Mr Clegg - said: "Nick Clegg should undoubtedly stay and the Liberal Democrats should stay the course in Government."
"There will be lots of people who are bruised by the results," he conceded.
He told BBC1's Andrew Marr Show: "I have lost elections before and it is miserable, and I also understand why these people will feel that the message that they got back on the doorstep, that they don't like us being in government and so on, is a really difficult one.
"But I just think, at this time, it would be absolutely foolish for us as a party to turn on ourselves.
"What has separated the Liberal Democrats and the Conservatives the last four years is that while the Conservatives have been like cats in a sack fighting each other, the Liberal Democrats have stood united. That is what we will continue to do."
Unlike the "spineless" other leaders, Mr Clegg had been "the only one with the bottle and the backbone to stand up to Nigel Farage", he told the Murnaghan programme on Sky News.
Mr Pugh told the Sunday Times a n internal "post-mortem" of the poor night at the polls - which saw the party almost or entirely wiped out in some former strongholds - "has to include a truly open, mature and balanced look at our whole strategy, including the leadership issue".
Colleague Adrian Sanders, who represents Torbay, said: "The problem is the messenger, very few people say it's the message."
Ex-MP Sandra Gidley - one of around 250 people to have signed an online letter demanding a change of leader - said Mr Clegg was tainted by his position in the Tory-led coalition.
"What we need over the next year is somebody to lead the party who is credible, who has questioned the coalition in the past, not seen to be in a love-in with David Cameron and has the confidence of the party," she told Murnaghan.
"We should be making more of a case for what we have done in the coalition and we are not able to do that at the moment while we have Nick, as a leader, associated with some of the negative aspects of the coalition - the broken promises.
"It's not easy to say this because Nick did a stonking job for us at the last election but I do not think he is the right face for the Liberal Democrats at the next election and many others sadly agree."
She backed Business Secretary Vince Cable to take over as an interim leader as he "has been shown to be independent-minded within the coalition".
Mr Clegg has conceded that the strongly pro-EU party could lose all 11 of its MEPS when results are declared at 10pm and even the most optimistic polls suggest only a handful will survive.
In Thursday's elections, the Tories took charge of Kingston Council - the back yard of Energy Secretary Ed Davey - and the Lib Dems lost control in Portsmouth following gains by Ukip.
As the scale of the losses became clear, Mr Clegg said he would "absolutely not" resign, and insisted the Lib Dems were still succeeding where they focused on their achievements in coalition.
Two would-be MPs have signed the online LibDems4Change campaign.
Jackie Porter, who is set to fight the Tory-held target seat of Winchester in next May's General Election, said the party was "not going forward with a clear strategy".
The county councillor said the party's achievements were overshadowed because Mr Clegg "allowed himself to be portrayed as just another pea out of the same pod" as David Cameron and Ed Miliband.
The open letter says: "We consider it vital that at the 2015 General Election the party should be led by someone who will receive a fair hearing about our achievements and ambitions for the future.
"It is clear to us that this person is not you, as the loss of so many of our hard-working councillors highlights."
If Mr Clegg refuses to quit, a leadership contest could be triggered if 75 local party associations formally demanded one, or if a majority of the parliamentary party approved a no-confidence motion.
Ros Kayes, the candidate in West Dorset, is another to put her name to the letter.
She told BBC Radio 5 Live: "If you were to ask the general public what it is they don't like, they wouldn't say they don't like the coalition, they would say they don't like Nick Clegg."
Mr Pugh told the Independent on Sunday that among backbenchers there was a " widespread view that no matter how much Nick spoke there was a section of the electorate that would not give him a fair hearing. But equally, there was a need to avoid any civil war in the party".
"We need to try to improve the national profile and try not to make Nick such an obvious target. It's the elephant in the room and the party can't ignore it," he said.
Another MP, John Hemming, said he would consider "whether any action is needed" when Parliament returned on June 4 but said he was " not an enthusiastic fan of Nick's leadership".
Former Lib Dem leader Lord Ashdown, who is chairing the party's 2015 general election campaign, said leadership change was "just about the silliest idea I have heard I think in my political career."
"It is just not serious politics," he told the BBC's Sunday Politics - quipping that a "Lib Dems Like Cake" website had more people signed up to it.
"This is the moment when we need to get out with a really good message and campaign through the summer in the context of the general election and we spend it on an internal, totally unnecessary, divisive leadership election? I think that's ridiculous.
"At the very moment when our sacrifices - and we have made some very big ones, painful ones - are beginning to gain traction, we turn in on ourselves."
Mr Clegg was "the best prime minister Britain hasn't got", he said, insisting that the message would get through to voters by next May.
The public had to be convinced that the party could "hack being in government".
"If we were to take this step the answer to that would be incontrovertibly 'no' and that would damage the party forever."
Mr Hemming said he would consult local party members over whether he should continue to support Mr Clegg but that he was personally "not of a view he must stay".
The organisers of the letter said that the signatories included 43 councillors, including two group leaders, three general election candidates and six local party chairs.
Former MSP Margaret Smith is also among those calling for a change as well as ex-c hair of the Association of Liberal Democrat Councillors Bill le Breton and county and city councillor Martin Tod - a member of the party's governing Federal Executive.
They rejected claims by Mr Farron that some of those on the list were not members - insisting that party membership numbers were required to register support.
"As the party president responsible for the activist base across the UK it is important that Tim Farron understands that this letter is entirely driven and supported by current members.
"We urge Nick Clegg to reflect on the contents of our letter and believe it would be remiss of any of our parliamentarians to ignore this group now."