Chairman Philip Day has told BBC Radio Humberside that Grimsby Town's next three games will not go ahead after a player tested positive for coronavirus.
The EFL has confirmed that Saturday's League Two game at Cheltenham is off.
Grimsby advised the league that they consider they are unable to safely fulfil the fixture.
But Day has said Grimsby's league game against Bradford on 3 October and their EFL Trophy tie at Hull three days later will also have to be postponed.
As a result of the positive test, other Grimsby players and staff have been required to self-isolate in line with EFL and government guidance.
The Lincolnshire club's training ground will now be shut down for 14 days.
Another of Saturday's league games - Bradford v Stevenage - is also in doubt after Stevenage reported one of their players had tested positive for the virus.
In a statement, Bradford said they are awaiting confirmation from the EFL about whether their games with Stevenage and Grimsby will go ahead as scheduled.
Grimsby 'put safety first'
Grimsby have also disclosed to the BBC that teenage midfielder Jock Curran was the player to test positive for Covid-19, and the 19-year-old is at home with mild symptoms.
"The rest of the squad and staff, who fill in a questionnaire and have their temperature taken every morning, have not displayed any symptoms of the virus," Day said.
Day says Grimsby took the decision to call off the game after consultation with EFL medical team and the club's own medical team.
"Our initial aim was to get everyone tested, and they should be okay as they are not showing any symptoms," he said.
"But the medical advice was that the incubation period is a minimum of five days, so those tests would be of no use whatsoever.
"We don't know if the players, having had contact with Jock on Monday and Tuesday, have the virus or are asymptomatic.
"We have put safety first and asked the EFL to call off the game."
Another EFL match postponed
Grimsby's trip to Cheltenham is the second EFL league fixture this season to be suspended because of Covid-19 cases.
Leyton Orient's game at Walsall in League Two on Saturday has also been postponed, as O's players and staff isolate after an outbreak of the virus at the club.
Orient's third-round Carabao Cup tie was called off on Tuesday after they received several positive tests, and opponents Tottenham Hotspur have been handed a bye into the next round.
An EFL statement said it would investigate the circumstances of the postponement of Grimsby's match.
A revised date for the fixture will be announced in due course.