Gov. Nathan Deal reiterated Tuesday that if the measure ends up on his desk, he'll sign it, so long as it allows communities to ask voters to decide.
"I believe that's what democracy is all about," Deal said. "I would not veto it if the Legislature passes it in a fashion that meets that criteria."
Georgia is one of just three states that ban the Sunday sale of alcohol at stores. The other two are Connecticut and Indiana.
Previous efforts to allow Sunday alcohol sales in Georgia stalled amid a veto threat from Deal's predecessor, Sonny Perdue, a Christian conservative who did not drink.
With Perdue gone, backers of Sunday sales say, its odds of passing have soared.
Senate Majority Leader Chip Rogers said the measure has "a greater chance of passing this year than ever before."
The bill, filed late Monday, would allow local communities to ask voters to decide whether they want to allow Sunday sales. Bars and restaurants in many parts of the state are already allowed to serve booze on Sundays.
Christian groups in Georgia are vowing to fight the measure, arguing increasing the availability of alcohol sullies the Sabbath.
Grocery and convenience store owners say their customers have been clamoring for the change.