Stevie Kilgower, the general secretary of Hearts Supporters Clubs, believes the demand of sitting on the club's board and managing the first-team set-up is proving too hard for Levein to handle.
Though not calling for Levein to quit the club altogether, he does believe the 54-year-old needs to focus on one aspect of his work at Tynecastle.
He told Sky Sports: "I don't know exactly how the dynamics work between the boardroom and the dressing room but I do wonder if it's all been too much for one man to take on.
"He's maybe spread himself a bit too thin. Craig, by all accounts, does a fantastic job at the academy and has turned that around with the youths we are bringing through.
"Maybe he spends too much time doing that on not concentrating on the first team. Maybe it's time to concentrate solely on the first team or move back upstairs and let somebody else do that."
Owner Ann Budge stated emphatically in a wide-ranging statement delivered on Wednesday evening that she remained fully behind her manager.
However, Kilgower believes the position will become untenable if Hearts drop their next two matches - away to Hibs in Sunday's Edinburgh derby and Aberdeen's visit to Gorgie next Wednesday for a Betfred Cup quarter-final encounter.
He added: "Probably before the last two home games, there was maybe a 50-50 split but home games this season have more or less cemented among the fans that it's not working for Craig.
"I do like Craig Levein and I've stuck by him myself over the years but the form in the last few home games has been shocking.
"There are two very important games coming up and it's unthinkable that we'll lose both. If we were to lose both, I don't see how things can continue in the current set-up."