The signing could well be the first for new manager Nuno Espirito Santo at the City Ground, and would see him link up with a fellow Portuguese compatriot.
Martins has been completely frozen out of the squad at Monaco by manager Adi Hutter, having not featured in almost a year.
The winger's last start came against Auxerre on February 1, and he has not played in any capacity since picking up an adductor injury later that month.
While Martins's pre-season was disrupted further by injury, his omission from the side in recent months has been at the discretion of manager Hutter.
With six months left on his contract in the principality, Portuguese outlet Maisfutebol claims that the player is in advanced negotiations to join Forest.
Instead of letting his contract run out and losing him on a free in the summer, Monaco would prefer to let the player leave in January in order to receive a transfer fee, even if it is only nominal.
After bursting onto the scene for Sporting Lisbon in the mid-to-late 2010s, Atletico Madrid spent €22m (£19m) to bring the wide man to the Metropolitano in the summer of 2018.
However, after failing to make an impact in the Spanish capital, Martins moved to Monaco on loan just six months later, and the Ligue 1 side made the move permanent the following summer.
Martins signed a five-year deal with the club, and enjoyed a productive first three years after making his move permanent.
However, injuries were disruptive even when he was a regular, and they have ultimately seen him fall further down the pecking order in the past two seasons.
At just 28 though, Martins still has plenty of his career ahead of him, and even though the report states that clubs in MLS and the Russian Premier League are interested in his services, the winger would prefer to test himself in the Premier League.
Forest took a gamble on another winger with a worrying injury history in Callum Hudson-Odoi during the summer window, and setbacks have disrupted his season, causing Forest to dip back into the market.
Keeping Martins fit could make it a very shrewd signing for the club though, given the quality he has shown in the past, especially in his final season at Sporting, where he contributed to 24 goals across domestic and European competitions.