Glasgow University said demand rose by almost 37% - to 6,339 applications.
It was a similar picture at other universities which traditionally attract students from England, Wales and Northern Ireland.
Edinburgh said applications increased by 24% - to 15,077.
With 2,665 applications, demand at Aberdeen was up by 26%.
St Andrews University reported an 11% rise.
It is believed demand was also up at Glasgow School of Art and at Strathclyde University.
The complete figures on applications for places at all UK institutions are due to be released by the admissions body UCAS later.
The upturn is all the more remarkable at Edinburgh and St Andrews which charge £9,000 each year - a total of £36,000 for the four year degree.Continue reading the main story
£9,000 for university education seems like a lot of money but comparatively is quite cheap”End Quote Alexander McNab Aberdeen University student In England, where degrees are generally over three years, graduates usually pay no more than £27,000 and also have one year less of living expenses such as rent and food.
Prof Ian Diamond, Principal of Aberdeen University, which charges a total of £27,000, said demand has gone up, not down, partly because of a strong marketing drive.
He confessed that when fees were launched this year he was not sure that demand would hold up.
He said: "I think we have all worried because we were taking a step into the unknown with no data on which to make predictions.
"We simply had to remain confident that the offer that we had at the University of Aberdeen was at least the equal of that of anywhere else."Healthy demand
Alexander McNab, from Suffolk, who is now a first year student at Aberdeen University suggested fees were not necessarily a barrier.
He said: "In many ways it is bargain because school fees are £16,000 - £26,000 in day and boarding schools. £9,000 for university education seems like a lot of money but comparatively is quite cheap."
The enviable international reputation of Scottish universities is another reason given for the healthy demand.
It is believed the provision of new bursaries to compensate for fees is also a factor - but there remains concern that students from less well-off homes may still be deterred from applying.
Early figures also seem to indicate more Scots are applying for university places. At Edinburgh demand was up 4.3%, at St Andrews it has risen by 4.5% and Glasgow saw a 10.2% increase.
And there are also signs of rising demand from EU students, who pay no fees in Scotland.