Football Insider claims the Ibrox side face competition from clubs in Germany and Turkey for the 29-year-old's services.
The future of Colombian striker Alfredo Morelos is unclear, with the 22-year-old attracting interest from a host of clubs after another goal-laden season, while Kyle Lafferty's position is also unclear.
The Northern Irish striker rejoined the Light Blues last summer after a spell at Hearts but despite scoring a double on his debut, struggled for game time and fell down the pecking order with Morelos and, latterly, on-loan Bournemouth forward Jermain Defoe preferred in attack.
Lafferty played 25 times for the Ibrox side but completed the full 90 minutes on just five occasions, with one coming in the Scottish Cup against Cowdenbeath.
Steven Gerrard is believed to be a fan of Iranian international Ansarifard, who has also had spells at Iran-based teams Saipa, Persepolis and Tractor Sazi as well as Osasuna in Spain and Panionios and Olympiacos in Greece.
He has scored 19 goals in 76 international appearances including one against Portugal in the 1-1 draw between the two sides at last year's World Cup.
Ansarifard is under contract at the City Ground until summer 2020 and while he has played just 12 game for Forest following a lengthy delay in obtaining a work permit and a spell at the 2019 AFC Asian Cup, his goal-scoring exploits for previous clubs have made him an attractive target for numerous clubs.