Rangers are yet to make a fresh approach for Jamie Walker and are unwilling to meet Hearts’ £1m asking price, putting any potential move before Thursday's deadline firmly in the balance, Sky sources understand.
The winger is out of contract next summer and is keen to move to Ibrox, but Hearts are, as it stands, refusing to negotiate below their valuation of the 24-year-old.
It is expected Rangers will make a renewed bid for Walker on Thursday, but the clubs are some distance apart from agreeing any deal.
New Hearts manager Craig Levein said at his unveiling on Tuesday that there had been no fresh offer from Rangers for Walker, who has refused to sign a new deal at Tynecastle.
Rangers had a bid for Walker turned down by Hearts earlier in the summer.
At the start of the season he was left out of the Hearts starting line-up by both former head coach Ian Cathro, and then interim head coach Jon Daly, who said the player's head "had been turned" by interest from the Gers.
If a deal for Walker fails to happen before the deadline, Rangers are confident they can secure a pre-contract arrangement with the player in January.