According to Fabrizio Romano, the offer is for a loan switch with a view to a permanent deal worth £20million next summer.
The 25-year-old has enjoyed hugely successful loan spells with Shrewsbury Town and Sheffield United, where he helped the Blades to secure promotion from the Championship before seeing them finish ninth in the Premier League in 2019/20.
Following such a hugely successful campaign in the top flight, Henderson opted to stay at Old Trafford in order to challenge David De Gea for the number one jersey, however claiming a starting spot has proved to be tricky after making just three appearances last term.
Ralf Rangnick confirmed back in January that Henderson had requested to leave the club, saying: "A couple of weeks ago he asked me [if he could leave] but I told him I'd rather have him stay until the end of the season."
Those comments came following links with a move to Newcastle, however reports of a move to Forest could be more appealing to both player and club.
Steve Cooper is looking to strengthen his side after securing promotion from the Championship, with Premier League experience high on his list.
With current number one Brice Samba facing an uncertain future after turning down a new deal at the City Ground, it seems that signing a new stopper is high on the club's agenda.
Other players linked with the club include Morgan Gibbs-White of Wolves, Keinan Davis from Aston Villa and Henderson's United teammate, James Garner.
Speculation over Henderson's future has been rumbling on for some time, with former United goalkeeper Mark Bosnich telling talkSPORT in October that Henderson should leave the club 'as soon as he possibly can' and that he needs to 'look after himself'.
"I think he should go. He's at the stage of his career where he needs to be playing regular football," Bosnich told talkSPORT Breakfast.
"He'll know that as well. I understand that at the start of the season, him having a go in pre-season to see how things plan out.
"De Gea has been excellent and there's no doubt about it, but I think from Dean Henderson's point of view, he needs to start looking after Dean Henderson.
"He should go and should go as soon as he possibly can."