The 22-year-old's future remains the subject of much speculation, with Forest facing a battle to keep hold of him before the end of the summer transfer window.
Aston Villa continue to be linked with the attacker, but according to The Telegraph, West Ham are now firmly in the hunt for his signature.
The report claims that David Moyes's side could move for the £45m-rated forward if Lucas Paqueta leaves for Premier League champions Manchester City.
Johnson scored 10 times for Forest during the 2022-23 campaign, including eight in the Premier League.
The forward also played 80 minutes of his side's 2-1 loss to Arsenal on the opening weekend of the new season.
Forest are in a strong position when it comes to any potential negotiations for the forward, as his contract at the City Ground is due to run for another three years.
Meanwhile, according to the Daily Mail, Celtic are keen to sign Wolverhampton Wanderers attacker Daniel Podence before the end of the summer window.
The 27-year-old made the move to Molineux from Olympiacos in the summer of 2019, and he has represented his current side on 105 occasions, scoring 16 goals and registering nine assists in the process.
Podence was a regular for Wolves last season, making 37 appearances, netting six times and proving one assist.
The Portuguese was not included in the squad for his team's Premier League opener against Manchester United on Monday evening, though, amid the speculation surrounding his future.
Celtic are said to be keeping a close eye on the forward, who is valued in the region of £12m.
Wolves are allegedly still planning to sell Podence this summer despite recently changing their manager, replacing Julen Lopetegui with Gary O'Neil.
Elsehere, Brighton & Hove Albion are allegedly interested in signing Boca Juniors defender Valentin Barco.
The 19-year-old is believed to have a release clause of around £8m inserted into his current deal.
According to The Argus, Brighton are huge admirers of the youngster and are planning an approach in the hope of bringing him to England before the end of the transfer window.
Barco, who is also capable of operating higher up the field, has been capped on two occasions by Argentina Under-20s.
The South American has a contract with Boca until next December, but the release clause will make it hard for the Argentine outfit to keep hold of him.
Barco has scored once and provided two assists in his 20 first-team appearances for the Blue and Golds.