The Englishman is out of contract at the end of the season and there has been little indication a renewal is forthcoming.
In turn, this month has seen Rangers linked heavily with centre-back duo John Souttar and Danilho Doekhi.
Both players, like Goldson, are into the final months of their deals with their current clubs.
Souttar is all but expected to leave Hearts, while Doekhi has suggested he will leave Vitesse Arnhem in the summer.
Pundit Charlie Nicholas reckons Giovanni van Bronckhorst would be willing to sell Goldson if one of the transfer targets arrived this month.
"He needs a Plan B if Connor Goldson decides to go this summer," the former Aberdeen, Celtic and Scotland star wrote in his Daily Express column.
"The Dutchman is also looking at Danilho Doekhi of Vitesse Arnhem, although there is interest from PSV Eindhoven.
"I believe if Rangers were to get one or both targets tied down this month, they'd look to sell Goldson in this transfer window."
Regarding Souttar, Nicholas reckons the player may want to leave Tynecastle before the close of the window with Rangers the most likely destination.
Hearts are very keen to keep him until the summer even if they are aware he is not going to sign an extension.
"If I was Souttar I would want to leave Hearts before the close of the window,” Nicholas said. “Hearts, though, can play hardball and let him run his contract down if the deal is not right.
"I can see the value in Rangers going for Souttar.
"Rangers have made a push and by all accounts Giovanni van Bronckhorst has made his move personally and I think that could have a big sway in his decision.
"I would think, as it stands, he will end up going to Rangers.
"The only way that might change is if Souttar gets an offer from an English Championship club where he has an opportunity to play and to win promotion to the Premier League.
"Souttar has had a lot of injuries but, at 25, he is still at a good age and if Rangers offer him a long-term contract then that could swing it."