Medway Council has issued a prohibition notice for the game after the Gills and Kent Police disagreed on the necessary levels of law enforcement on matchday.
The same action was taken ahead of Gillingham's FA Cup third round tie against Stoke in January, only for the relevant parties to resolve their differences and allow supporters to watch the game.
Councillor Mike O'Brien, portfolio holder for community safety, said in a statement: "Due to a dispute between Gillingham FC and Kent Police about the costs and number of police officers needed at Priestfield Stadium for the match against Swindon Town on April 21, we have unfortunately been left with no alternative but to issue a prohibition notice on the game preventing supporters from attending.
"We have done this following advice from Kent Police that they have specific intelligence concerning the match and its fans.
"We are satisfied that the intelligence provided shows the police are clearly right to have serious concerns about public order at the ground and that the risk is so serious for fans that, until steps have been taken to reduce it to a reasonable level through agreeing adequate policing levels, spectators will be prohibited from attending the match."