The former Wigan and Chesterfield forward had a clause in his contract allowing him a new deal if Hull won promotion and now the club are ready to tie down Folan.
"Caleb is close to signing a new deal," Hayden Evans told skysports. "He is entitled to a new deal because of a clause in his old contract.
"We have started dialogue and there is a willingness from both parties to do it.
"I hope we can get things sorted out very soon."