A second bid in the region of £7million, according to Press Association, comes less than a week after the Premier League club saw their initial approach turned down.
The Hammers had offered around £5million plus Matt Jarvis on a season-long loan deal.
Forest had also previously rejected bids from West Brom and east midlands rivals Derby for Antonio, who is it understood is due in London for talks on Monday.
Antonio has made a huge impact at the City Ground since joining Forest from Sheffield Wednesday last summer in a £1.5million deal.
He scored 15 goals during his debut season on the banks of the River Trent and has already netted four this campaign – with his goal on Saturday in the 2-1 home defeat to Cardiff seemingly his last for the club.
The 25-year-old has been keen to move and was left out of the Forest squad recently by manager Dougie Freedman following the bid from Derby, with Freedman insisting he was not in the right frame of mind to play.
Should the deal for Antonio go through it could have huge implications for Forest next summer as it could see the club’s transfer embargo for breaching Financial Fair Play rules lifted.
More immediately, it would free up a space in the squad for Freedman to sign a player either on a free transfer or on loan ahead of Tuesday night’s deadline.