The alcohol ban – which does not cover hospitality sections of grounds – was introduced following a riot during the 1980 Scottish Cup final between Rangers and Celtic at Hampden.
The Scottish Beer and Pub Association (SBPA) argues the restrictions are outdated and it would like to see a season-long trial being held, paving the way for a possible permanent lifting of the ban.
Trials have recently been held at matches at Arbroath, Ayr United, Queen of the South and Stirling Albion prior to the start of the league season.
St Johnstone was also granted permission to sell alcohol to supporters at two games but the Pittodrie trial was the first for a top-flight match.
A legislative change would be required to amend part of the Criminal Law (Consolidation) (Scotland) Act 1995 to allow the regular sale of alcohol at football stadiums.
A Scottish government spokesperson said: “Our position on alcohol sales at Scottish football grounds has not changed.
“It is for local licensing authorities and Police Scotland to satisfy themselves that the pilot projects are permissible within the current legislative framework when granting licences.”