Labor and independent senator Jacqui Lambie want the minister to go.
Opposition Higher Education Spokesman Kim Carr says the minister's relationship with the Senate is now poisonous, after the upper house rejected his legislation 34-30 last night.
Just three of the crossbenchers voted with the government when it needed support from six of them.
Pyne insists he'll bring the reforms back to parliament for a third time, probably in June.
"This minister's bluff, his bluster, his bravado simply leaves people cold," Senator Carr told ABC radio on Wednesday.
"It is time for him to consider his future and if he doesn't then he should be sacked."
Senator Lambie said Pyne - who's labelled himself a "fixer" on education policy - had failed despite using everything from blackmail to pushing university vice-chancellors into a corner.
But Prime Minister Tony Abbott is standing by his minister.
"He's doing a really, really good job. Of course he is," he told Fairfax Radio.
Of course he has to say that. Pyne is one of his key supporters. The way they have trashed South Australia ( destroyed the car and ship building industries there) it will be a miracle if he survives beyond the next election anyway.
Government Senate leader Eric Abetz lamented there was undue concentration on how the crossbenchers voted when the focus should have been on Labor and the Greens.
Labor was arrogant to think it knew better than vice-chancellors what was good for the tertiary sector, he said. Senator Abetz calling people arrogant! That's the pot calling the kettle black!
Greens leader Christine Milne said the crash-through-or-crash approach just showed the government wasn't learning from its failure to convince the Senate.
And Labor piled on, with MP Ed Husic saying the reforms clearly were just in suspended animation.
"It's definitely the Frankenstein bill, it's got bolts in the neck," he said.
"It will be scaring children in years to come."Senator Lambie said there had been enough talk.
"We have been listening. That's why the deregulation hasn't got through."
Extract from an AAP story.