Specialists have informed the Bosnian that he does not require an operation after aborting a return to light training last month due to persistent pain. He sustained the original injury shortly after pre-season training began in July, however, the length of recovery time is frustrating club officials.
Beslija must now undergo an intense fitness programme having missed the whole of Hearts' pre-season preparations.
He arrived at Tynecastle in January 2006 for a club-record 800,000 from Racing Genk but continual injuries and illness have restricted him to just 13 appearances in maroon.
Manager Csaba Laszlo conceded today that it may be some time before Beslija is challenging for a first-team place.
"We cannot use his services at the moment. We don't see a big development," said Laszlo of the winger's recovery. "At the moment, I am thinking about the players which are not injured and the ones which can come back quickly. I am waiting for Bruno Aguiar and Mike Tullberg.
"If a player comes back who has not trained then we must push him to be available for 90 minutes, but this is not easy and takes time. The priority is to keep the team together and get our injured players back."
Zoran Krneta, a director of the Star Management Signings agency which represents Beslija, told the Evening News: "He is trying to get better, hopefully it will be about a month or maybe more. I think it will be a month minimum.
"He was supposed to go to London last week but didn't. I spoke to Hearts a few days ago and, obviously, they are not happy but it's not really anybody's fault. Mirsad certainly doesn't want to be injured.
"He doesn't have a brace any more and the specialist told him he doesn't need an operation. I spoke to Mirsad and he just wants to be back playing."
Tullberg has returned to Denmark for treatment on his muscle rupture and should be back in Edinburgh within a week. Laszlo continued: "He will be back in one week and he should be able to train with the team."