The Premier League team was rumored to be interested in signing the Croatian international over the summer, since his contract at Ibrox expires in the summer.
Barisic may leave the Rangers at the end of his current contract, despite having played in almost 200 games since signing from Osijek in 2018.
That is in spite of rumors that the 30-year-old would remain with the team for an extra two years if Rangers were to provide a new contract.
Renan Lodi chose to join Marseille in Ligue 1 rather than make his loan move permanent to the City Ground, leaving Forest in need of a left-back during the summer.
Even though they may agree on terms on a free transfer in January, the Daily Record believes Forest is no longer interested with the acquisition of the currently injured Nuno Tavares.
Even with the recent acquisitions of Tavares and Ola Aina from Torino, Forest is still somewhat of a crisis at that position.
The two have missed the last two Forest games against Brentford and Crystal Palace. Harry Toffolo, who did not play in the opening two months of the season, has been brought back into the starting lineup.
Even though Aina is right-footed, she was covering at left-back prior to Tavares’ arrival, but the injury to the loanee from Arsenal has further limited Steve Cooper’s choices.
Cooper suffered another setback at the full-back position earlier in the week when Gonzalo Montiel hobbled off while playing Argentina due to a potential muscular problem. However, Neco Williams has now returned to full fitness.
Barisic may be ready to return to his native Croatia, as Dinamo Zagreb is rumored to be interested, if Forest decides to pass on him.
Although Barisic is still a frequent starter, his future is uncertain given the Rangers’ declining performance and the arrival of new manager Philippe Clement.
Earlier this week, Croatian publication Novometne Vijesti said that Dinamo and Forest were both considering acquiring Barisic on a free transfer.
But according to the Daily Record, Rangers will want to work out a new deal with the player so they can get paid a transfer fee should they decide to sell in the winter transfer window.