The Uruguay international requested that the fine he suffered for biting the arm of Chelsea defender Branislav Ivanovic go towards the families, who lost loved ones in the disaster of the 1989 tragedy.
While Aspinall has appreciated that it is a "nice gesture", the justice campaigner is torn over whether to accept it due to Suarez's actions following the 24th anniversary of Hillsborough.
"I would rather not be accepting the fine from Luis Suarez because I would rather he had not bitten that player," the Daily Mail quotes Aspinall as saying. "It is a nice gesture but it is a terrible thing under the circumstances.
"We can't say we are grateful because we would rather him not have done that at all. It is an awful thing at any time, not just this particular week.
"It is nice to know he knows he has let himself, the club, the fans and the [Hillsborough] families down. We could refuse to accept it but it is going to have to go to someone, and he wants to show respect and remorse to the families."
Meanwhile, Suarez is expected to be hit with a lengthy ban by the Football Association this week for his controversial actions.