Pressure is growing on the Scottish Labour leader Jim Murphy after the Unite union called on him to resign.
The union, which was the party's biggest donor during the last parliament, said Mr Murphy should stand down "without delay".
It warned that otherwise "extinction looms" for the Scottish party.
Aslef, the train drivers' union, also called for Mr Murphy, who lost his East Renfrewshire seat in Thursday's poll, to resign his position as leader.
Labour lost all but one of its 40 Scottish seats to the SNP.
On Friday, Mr Murphy said he would remain leader of the party in Scotland and stand for a seat at Holyrood in next year's Scottish parliamentary election.
In a statement, Pat Rafferty, leader of Unite in Scotland, said: "It is time for change at the Scottish Labour Party.
"That is the overwhelming, unambiguous message from Scotland's people, including its trade union members, on Thursday.
"Either Scottish Labour rediscovers its mission as the natural voice for social justice in our nation, or irrelevance and ultimately extinction looms."
Mr Rafferty added: "Change must begin with a new leader. It is surprising that Jim Murphy should feel he still has a mandate to lead the party after Thursday's results.
"I call on Jim Murphy to resign without delay, and give the membership of the Scottish Labour Party the chance to determine their own way forward in rebuilding from Thursday's ruins."
In response, a spokesman for Scottish Labour said: "This is no surprise. They [Unite] wanted him to resign before he even became leader."
However Kevin Lindsay, Aslef's organiser in Scotland, said: "Jim Murphy has just presided over the worst election defeat in the history of the Scottish Labour Party.
"He has to go - and he has to go now."
'Radical solutions needed'
Elsewhere, Labour MSP Neil Findlay MSP, who stood against Mr Murphy for the leadership of the party, has resigned from the Scottish shadow cabinet.
In a statement, he said Thursday's election had been a "disaster" for Scottish party.
He added: "The problems are wide-ranging and deep.
"Radical solutions are needed and can only be implemented following a full, frank, open and democratic debate led by our loyal and hard working, committed party members - a centralised fix just won't do."
Earlier in the day, former Scottish Labour minister Andy Kerr had said his party needed to apologise for an "atrocious campaign".
Mr Kerr, who failed to win the Scottish Labour leadership in 2007, said the party in Scotland needed to be independent of the UK organisation.
He compared Mr Murphy's campaign to "dad-dancing in the most modern nightclub in the city of Glasgow".