The Conservatives say they will write to the 42 senior football clubs seeking their views on selling alcohol at matches.
Alcohol can be consumed by ordinary fans in England before matches and at half-time inside football stadiums, as well as at other sporting events in Scotland.
Drink was banned at football games because of rioting during the Scottish Cup final in 1980, but the Conservatives believe it is time to look again at the restriction.
Alcohol can be consumed in corporate hospitality areas when the match is not in play, but drink is not permitted anywhere else in grounds in Scotland.
Miss Davidson said: "I understand why the ban was brought in all those years ago, but times have changed significantly since then. We now have modern stadia with excellent stewarding to keep spectators safer, allowing more families to attend matches. This has resulted in the football watching experience being a far more civilised experience than it was 30 years ago.
"With that in mind, it's time to consider allowing supporters to enjoy an alcoholic drink before the game and at half-time. There's no reason why this should be successfully in place in England but not north of the border.
"It seems unfair that people who can have a drink in a bar, rugby ground or even at a concert in football stadiums are unable to consume alcohol before they sit down to watch a game of football. Such a change would have the potential for our cash-strapped clubs to bring in some extra income, while boosting the match-day experience for thousands of fans every week."