Adams's deal with Southampton is due to expire at the end of the month, and despite helping the Saints secure a return to the Premier League for the 2024-25 campaign, he is being tipped to move on.
Wolves were thought to be at the head of the queue for the Scotland international, who scored 17 goals and registered six assists in 46 appearances for Southampton last season.
However, according to talkSPORT, Forest have now moved into pole position, with the Reds confident that they will be able to secure a deal for the forward.
Adams, 27, has scored 48 times and registered 20 assists in 191 matches for Southampton.
Meanwhile, he is experienced in the Premier League, turning out on 124 occasions in England's top flight, scoring 25 goals and registering 13 assists in the process.
Elsewhere, according to Football Insider, Everton are planning a move for Sheffield United defender Max Lowe, with the 27-year-old set to be available on a free transfer.
Lowe's contract with the relegated Blades is due to expire at the end of June, and a number of clubs are believed to be showing an interest in his services.
Everton are said to be admirers of the Englishman, who featured on 10 occasions in the Premier League during the 2023-24 campaign.
Lowe struggled with injury last term, but he did return for Sheffield United's final two league matches against Everton and Tottenham Hotspur.
Toffees head coach Sean Dyche allegedly views Lowe as the ideal player to strengthen his defensive options for the 2024-25 season.
Meanwhile, Celtic are reportedly considering making a move for Bournemouth defender Chris Mepham during this summer's transfer window.
The 26-year-old only made 13 appearances for the Cherries during the 2023-24 campaign, and he missed the end of the season due to health problems.
The Wales international has another 12 months left to run on his current deal, but he is being tipped to leave the Vitality Stadium in the upcoming market.
According to the Daily Mail, Celtic are interested in the defender, who is believed to be valued by his current side in the region of £8m.
The report claims that Leeds United and Sheffield United had been keen to sign Mepham in January, but Bournemouth refused to sell in the middle of the campaign.
Celtic head coach Brendan Rodgers is said to be determined to boost his defensive options this summer.