Hull City chairman Assem Allam will refund season ticket holders who wish to return their tickets in protest at the club's proposed name change.
Egypt-born businessman Allam's plan to rebrand the club as Hull Tigers has been criticised by some supporters.
Fans' group City Til We Die has opposed the move and hopes the Football Association will block the name change.
A Hull statement said Allam made the cash-back offer after meeting with supporters to discuss the proposal.
"Dr Allam has made this offer in light of the confusion that followed the recent meeting with supporters' groups and the fragmented messages that have emanated since that meeting," it said.
"That meeting closed with one of the representatives requesting the opportunity to develop alternative commercial models and nothing has yet been presented to the club."
Allam insists the 'Tigers' brand would make the club more marketable and said he considered the word 'City' to be "lousy" and "common".
The 74-year-old, who moved to Hull in 1968, said he had to find other income streams for the newly promoted team as their stadium is owned by the council.
"He is also keen that the majority of fans are able to focus on enjoying the quality of Premier League football at the KC Stadium, without being diverted onto minority interests," the statement added.
Steve Bruce's side are 12th in the table, having won promotion back to the top flight last season.