The Croatian international is out of contract at Ibrox next summer, and the Premier League club were believed to be interested in his services over the summer.
Barisic has made over 200 appearances for Rangers since joining from Osijek in 2018, but may depart at the end of his current deal.
That is despite reports claiming Rangers have tabled a new contract which would keep the 30-year-old at the club for an additional two years.
Forest were in the market for a left-back over the summer after Renan Lodi opted against making his loan move to the City Ground permanent, and instead joined Marseille in Ligue 1.
After the arrival of the currently injured Nuno Tavares though, the Daily Record claims Forest are no longer interested, even though they could agree terms on a free transfer in January.
Forest are still facing something of a crisis in that position despite the recent acquisitions of both Tavares and Ola Aina from Torino.
The pair have both missed the past two Forest matches against Brentford and Crystal Palace, with Harry Toffolo being thrust back into the fold after not featuring in the first two months of the season.
Aina, while right-footed, had been filling in at left-back before Tavares arrived, but the injury to the Arsenal loanee has stretched Steve Cooper's options even further.
While Neco Williams has returned to full fitness, Cooper was dealt another blow in the full-back area earlier in the week, when Gonzalo Montiel limped off while representing Argentina with a possible muscular issue.
With Forest set to snub Barisic though, the Croatian could be set for a move back to his homeland instead, with Dinamo Zagreb believed to be interested.
Barisic remains a regular starter but with Rangers' form dipping and new manager Philippe Clement recently coming through the door, his future is up in the air.
Croatian outlet Nogometne Vijesti reported that both Dinamo and Forest were interested earlier this week, especially with the prospect of getting Barisic on a free transfer.
However, the Daily Record believes that Rangers will hope to agree a new contract with the player so they can receive a transfer fee if they were to sell during the winter window.