The Football Association council will meet on Wednesday to decide on the proposal, even though the membership committe has already made a recommendation to reject the plans.
A total of 2,565 of the fans voted for the change, to 2,517 against.
Fans' group City Till We Die said the result was "largely meaningless".
Of the club's 15,033 season card holders, 5,874 responded, with 792 voting they were "not too concerned and will continue to support the club either way".
Those in favour voted "Yes to Hull Tigers with the Allam family continuing to lead the club" while those against voted "No to Hull Tigers".
In a statement, City Til We Die said: "The truth of the matter is that this result is largely meaningless, either as a true measure of opinions regarding the name change proposal, or the supporters' desire to see Dr Allam and his son Ehab stay."
Hull are currently 12th in the Premier League and have a chance to reach the FA Cup final for the first time on Sunday if they beat League One side Sheffield United in the clubs first semi-final for 84 years.