The Education Secretary is expected to update MSPs on his plans this coming week, amid growing calls for a decision to be made on the issue before Christmas.
The National 5s have already been cancelled next year, but the Scottish Government initially planned to press ahead with the Highers and Advanced Highers.
But there are growing concerns that many children will have lost out on schooling after being told self-isolate - and that the situation is not fairly spread across the country.
Tory education spokesman Jamie Greene said: “Pupils, parents and teachers need urgent clarification over what is happening with Higher exams. It is unacceptable that they continue to be left in limbo as another week starts.
“Yet again John Swinney is failing to show leadership when it matters most, repeating his dithering and delaying tactics which we’ve had to become accustomed to.
“This constant indecision is exactly why John Swinney has completely lost the faith of those involved in our education system.”
The Tories will stage a debate at Holyrood on next year’s exams diet in a bid to secure an announcement from Mr Swinney before the Christmas break.
The exams diet has been cancelled in full in Wales next summer and opposition parties want Scotland to follow this example.
Liberal Democrat leader Willie Rennie said: "With plenty of warning, it is possible to put alternative methods of assessment in place. It will be much harder to do that if ministers kick the can into the new year or create more work for teachers over the Christmas holidays.
"Teachers and parents are concluding it wouldn't be fair to go ahead with nationwide exams as normal.
"The Scottish Government needs to end the uncertainty, immediately announce that the exams won't go ahead and establish the credible alternative before Christmas."
There are concerns that some pupils will not have had any disruption this school year, but others will have had to self-isolate multiple times. Work is ongoing with local councils to gauge the level of disruption in each school.
The Education Secretary said last week there were differences in terms of geography, with the virus being more prevalent in certain areas than others.
A Scottish Government spokesman said: “Our ambition remains for Higher and Advanced Higher exams to proceed if possible.
"It must, however, be safe to do so and it must be fair to learners. We are monitoring the position closely and listening to all the different views on this matter.
"We have already shown with the cancellation of the National 5 exams that we will take the necessary actions to protect fairness for learners.
“The Deputy First Minister has been clear that the latest point at which a decision will be taken is the February break, which is more than three months before the exams are due to begin. He has also been clear that he will make a decision sooner if the evidence is pointing to that.
“Contingency plans are being drawn up to ensure fair and consistent awarding for Higher and Advanced Higher courses should exams not be possible.”