The club say Challinor – who memorably led Pools back to the Football League in June - has asked to be released just days after signing a new three-year contract.
He is set to join former club Stockport County in a move that would see him return to non-league football.
Hartlepool United Supporters’ Association secretary Paul McSweenie said the news was “very, very disappointing”.
She suspected there was more to Challinor’s sudden departure then met the eye, adding: “It seems pointless to sign a new contract and then just walk away.
"The fact is he has stepped back down to being a non-league manager. It just does not seem right.”
She feels the club should move swiftly to appoint a new boss in order to maintain the current momentum – with Pools enjoying a top-half position - saying: “He will be a big miss but we just need to move on.
"And we certainly need to block him from coming back for our players. We can't let him come here and take what we have got.”
Ron Harnish, of the Hartlepool United Supporters Trust, said fans were understandably upset.
"It is devastating news but unfortunately, that is football,” he said. "It is just one of those things – football managers do move on to what will, hopefully for them, be a better place.”
He said It was unsurprising that Challinor had attracted attention during his successful time with Hartlepool and continued: “Dave has proved himself with us – he pulled us up by our bootstraps two years ago.
Ron did not blame the manager for finding the prospect of a move back to his native North West attractive, adding: “At the end of the day, he has done what is best for himself and his family.”
Neil Appleyard, of the club’s disabled supporters’ association, said: “It seems strange that we are in the Football League and Dave has decided to go back into the Conference.
"But if he wants to be closer to where he lives, closer to home, that is understandable.
"I am shocked by the whole situation, to be honest, but at the end of the day, it is what it is."