A number of top sides have backed calls for a pilot scheme allowing the sale of drink to be introduced to see if the existing ban could be lifted.
Almost 90 per cent of clubs – including several top-flight teams – told the Conservatives they support a trial.
But First Minister Alex Salmond says Police Scotland are currently “not minded” for the ban to be relaxed.
A total of 15 of Scotland’s 42 Scottish league clubs back a trial, with two opposed, a poll found. The responses were secured under condition of strict anonymity, but are believed to involve a number of the country’s biggest clubs.
It follows a study earlier this week which found that two-thirds of Scottish fans want to see the alcohol ban lifted.
Scottish Conservative Party leader Ruth Davidson said: “Supervised, responsible drinking happens in other sports in Scotland and with football in other countries. Are we really saying that Scottish football fans cannot be trusted?
“I’ve written to every senior club in Scotland to canvass opinion, as well as to the Chief Constable of Police Scotland to discuss any security concerns.
“Of the clubs that responded to our survey, 88 per cent were in favour of a pilot project to see if the ban can be lifted.”
The alcohol ban was introduced at all stadia across Scotland after violent scenes marred the 1980 Scottish Cup Final when Rangers and Celtic fans fought running battles on the pitch at Hampden after the match and mounted police helped restore order.
The Scottish Government last looked at the issue in football seven years ago. But justice secretary Kenny MacAskill ruled out lifting the ban after Rangers fans rioted in Manchester following the side’s Uefa Cup Final defeat in 2008.
Mr Salmond said yesterday that decisions on this were “informed by Police Scotland, who confirm that they at this stage are not minded to seek a relaxation of the controls on alcohol at football matches, but are engaging with interested parties on this matter”.
But he said: “We have to have an approach which understands alcohol is a major contribution to disorder in society and to disorder and offensive behaviour at football matches.”
Rugby fans are allowed to drink after the same ban was relaxed at Murrayfield in 2007.
Lifting the drink ban could help cash-strapped clubs earn some extra income, while boosting the match-day experience of tens of thousands every week, backers claim.
Ms Davidson insisted that having “properly supervised alcohol” on sale inside football grounds “could be an important new revenue stream for Scottish clubs”. She added: “All-seater grounds and extra measures address some of the safety fears of the past, when this ban was brought in.
“Why not try a pilot project to see if what we know works well elsewhere can work here too? The fans want it, the clubs want it and it is in everybody’s interest to make any relaxation of the rules work.”
SNP back-bencher Jim Eadie also backed calls for the alcohol ban to be relaxed. “Scotland has moved on significantly since the alcohol ban was imposed almost 30 years ago, with all-seater stadiums,” he added.
“Isn’t it time then that we reviewed the ban and would it not be possible to lift it on a trial basis?”