Surridge joined the club from Stoke City in January 2022, and scored seven goals as Forest sealed promotion to the Premier League after over two decades away.
However, having fallen down the pecking order at the City Ground and with just one year left on his current contract, Forest are set to move the 24-year-old on.
The Athletic claims that a deal has been struck which will see Surridge move to the States for a figure of £5m.
Potential performance and appearance-related add-ons means that the deal could eventually rise to around £6m, which would be a club record for one of Major League Soccer's newer franchises.
The initial fee is just short of Nashville's current record of £5.2m, which they spent on Ivorian forward Ake Loba from Monterrey in 2021.
Surridge will hope to have a much greater impact at the club than the man he is essentially replacing though.
Loba now finds himself out on loan back in Mexico at Mazatlan, the side which finished bottom of the previous Liga MX Clausura, having scored just two MLS goals in 40 appearances.
It is believed Surridge had offers from Championship clubs, but prefers a change of scenery for his next career move.
Surridge featured 20 times in the Premier League last season, with the vast majority being as a substitute, as new signing Taiwo Awoniyi was the preferred choice up top.
The former Bournemouth and Swansea City man only managed one goal across the league season, but he did manage a further two when given more game time throughout Forest's run to the EFL Cup semi-finals.
Forest have also already confirmed the permanent signing of Chris Wood from Newcastle United after his six-month loan spell last season, adding further competition for places at the City Ground.
Nashville are currently third in the Eastern Conference, and Surridge could end up playing alongside one of the league's finest players, Hany Mukhtar, who has contributed to 21 goals in 23 MLS matches so far this season.