Education Secretary John Swinney could face a no-confidence vote at Holyrood over the exam grades controversy.
Scottish Labour said it would table a motion calling for his resignation after a suggestion some pupils may not get appeal results until the end of May 2021.
The Scottish Conservatives said they would back the motion.
Mr Swinney has rejected claims that pupils from less well-off areas were discriminated against.
The Scottish government has faced criticism over the grading system that replaced the 2020 exams which were cancelled because of the coronavirus pandemic.
The system, implemented by the Scottish Qualifications Authority (SQA) and approved by the government, asked for teacher estimates of students’ performance.
But more than a quarter of grades were changed during the moderation process based on criteria that included schools’ historic performances - with a total of 124,564 pupils’ results downgraded.
About 100 pupils took to Glasgow’s George Square on Friday morning to protest against the awards system - the methodology of which was only revealed on results day - claiming it discriminated against pupils in poorer areas.
The first minister said the process was “effectively statistical moderation” and argued results would not have been “credible” if the pass rate of the most-deprived pupils had risen by the 19.8% estimated by teachers before moderation.
Both Nicola Sturgeon and Mr Swinney have said the free-of-charge appeals process would allow eligible pupils to challenge their downgraded results.
But Scottish Labour says it has a photo of the SQA’s intranet suggesting a potential nine-month wait for pupils who had appealed but were not awaiting college or university places.
According to Scottish Labour, the 31 May 2021 date for appeal outcomes to be released has now been removed from the exam board’s internal website.
However, the SQA has since strongly denied the result of appeals will not be known until May.
The result of urgent appeals - those from students hoping to go to college or university - will be known by 4 September.
It is expected the date for the result of other appeals will be known in about two weeks with the likelihood it will be no later than Christmas.
The deadline for submitting appeals for “priority” students is 14 August and 21 August for all other pupils.
‘Shambles upon shambles’
Calling for Mr Swinney’s resignation, Scottish Labour’s education spokesman Iain Gray said: “Since the shambles of the SQA results emerged on Tuesday, the SQA and SNP ministers have deflected criticism through arguing that students could appeal unfair grades.
“This astonishing leak blows the lid off their defence.
“The SQA created this mess and the SNP government has entrusted them to sort it out - but all we have seen is shambles upon shambles upon shambles.
“To throw young people’s life chances into doubt is a disgrace, but to then make them wait over nine months for justice is a total insult.”
He added: “It is now clear that John Swinney has completely lost control of the SQA and the exam process, and he needs to go.
“We will seek to lay a motion to that effect and approach colleagues across parliament for their support.”
Under Scottish Parliament rules, a motion of no confidence can be tabled, debated and voted on if it is backed by at least 25 MSPs.
Asked earlier whether Mr Swinney or the SQA’s chief examining officer Fiona Robertson should “take responsibility” and resign, the first minister said there were “debates to be had” once the results process was over and she hoped the free appeal system would address individual anomalies.
Speaking at the government’s coronavirus briefing, Ms Sturgeon said: “If we get a situation where lots of appeals are awarded, then it will show that that process has worked as intended. So let’s wait until we get to the end of that process and see what the situation is [then].
“There will be questions asked, there will be debates to be had, there will be a reflection the government and SQA want to make about the method used… I’m not suggesting otherwise.
“But don’t just cast aside this next part of the process because this is the part of the process that we always intended would be about ensuring the individual injustices could be identified and rectified.”
Scottish Conservative education spokesperson Jamie Greene said his party would support the proposed no-confidence motion.
“Public confidence in Mr Swinney’s ability to handle his brief, particularly his response to the current SQA fiasco, has hit rock bottom,” he said.
“Reports of the SQA appeals process potentially being extended to May 2021 will be gut-wrenching to young people waiting on a training or university place.”
The Scottish Liberal Democrats said they were waiting to hear more from John Swinney on how to address the complaints, but did not rule out backing a no-confidence motion.
Scottish Lib Dem leader Willie Rennie said: “”John Swinney and the SNP government have let down thousands of students just at a time when they needed them most. The have baked in the inequalities in education that have been growing for over a decade.
“For the sake of the students we need to focus on resolving this issue and therefore we want to hear from the education secretary on Tuesday about his plans. If we are not satisfied that he has a plan that will work we will consider backing a motion of no confidence.”
The Scottish Greens said they did not yet have a position on the no confidence vote but are calling for Mr Swinney to adopt a “no detriment” policy for the appeals process which would see students getting no lower grade than had been achieved in their prelim.
Education spokesman Ross Greer said: “The Scottish government’s handling of the entire SQA fiasco has been disastrous. It has so far failed to acknowledge there’s even a problem, let alone offer any solutions.”