Flanagan sustained the injury during training on Friday and is heading back to parent club MK Dons.
He is likely to need an operation and will be out for a minimum of two months. His half-season loan from MK Dons will have expired by then.
Gills boss Martin Allen told website Kent Online: "I fully expect by Thursday there will be further strengthening of our squad. I am looking to strengthen the team and we have quite simply got to get better."
Allen said Gillingham may look to bring Flanagan back once he recovers. "We will see," Allen said. "I don't know. It's up to MK, it's not up to us."