Payne suffered the injury in Saturday's defeat to Shrewsbury following a challenge with Marvin Morgan and he later left Priestfield on crutches.
Gills boss Andy Hessenthaler may also be robbed of second-choice goalkeeper Paulo Gazzaniga for four weeks if doctors confirm he has come down with Mumps.
Gazzaniga reported for training on Monday with swollen glands and has been told to stay away from his team-mates as a precaution.
Hessenthaler may use defender Matt Fish as a back-up option because Daren Hawkes cannot be recalled from his loan spell with Ramsgate and youth team stopper Ashley Bourne is injured.
"Funnily enough Matt Fish is a really good keeper so he might be the answer. I've seen him play there a few times in training, when we were at Dover, and he's not afraid to throw himself around a bit," Hessenthaler told the Gills' official website.