The winger suffered a hamstring injury while training for last Thursday's Europa League qualifying match against Maribor.
The 29-year-old Irishman has had a scan on the injury, and now Dons boss Derek McInnes does not expect to have Hayes available until September.
McInnes said: "[Hayes] will probably be out for up to four weeks. So I think we would rule him out until after the international break.
"In saying that, he's been a quick healer in the past. Jonny's four weeks has sometimes been three weeks, so we live in hope with that. But we've got to be careful with a hamstring."