Medway Council claims the club owed more than £34,000 in unpaid utility bills and charges for using Beechings Cross.
The council said Gillingham have been using the facility "without permission" and considered the club to be trespassing.
They have indicated they will begin legal proceedings if the Gills do not hand back the keys to the ground by Friday.
"The money owed is not a small figure, and it is public money provided by the taxpayer. It would not be right, or fair to taxpayers or those that pay their rent, to ignore this," Alan Jarratt, deputy leader of Medway Council, told BBC Sport.
"We have tried as hard as we can to negotiate with the club and find a way through, but they have failed to talk to us. It is a shame it has come to this."