The messages appeared on the account of Andy Bell, a vice-president of Carlisle United, on Tuesday, but were deleted shortly after.
The club said it was "shocked" by the "disrespectful" comments and added it in "no way supported or condoned" them.
Mr Bell denied responsibility saying his Twitter account had been hacked and apologised.
The initial tweet read: "Don't mention the Heysel Stadium disaster!" - referencing the deaths of 39 people in Belgium in 1985 in the build-up to the European Cup final between Liverpool and Juventus.
It was followed by another which said: "Were you involved in a stadium disaster that wasn't your fault? Call us now and get the compo you deserve!"
The tweets were published on the day an inquest jury declared 96 Liverpool fans were unlawfully killed at Sheffield's Hillborough Stadium in April 1989.
In a statement, Mr Bell apologised for any offence caused and said someone had tried to "tarnish" and "slur" his name.
He said: "Nevertheless, I take full responsibility for failing to make sure [my account] was correctly secured to avoid being hacked."
The statement added he has asked Twitter to investigate how his account was targeted.
Carlisle United Official Supporters' Club described it as a "sick incident" and called on Mr Bell to provide evidence he had been hacked.
A football club official has come under fire over "sick" tweets mocking victims of the Hillsborough tragedy.