The St Etienne winger has reportedly rejected offers from former club Leeds United and Hull City in favour of a move to Upton Park.
Gradel moved to St Etienne from Leeds in 2011 after refusing to sign a new contract but he has struggled to cement a first-team place, making just nine league appearances in each of the last two campaigns.
The 26-year-old impressed during his time at Elland Road. Having initially joined on loan from Leicester City in 2009, he helped the club to promotion from League One and scored 18 goals in the Championship the following season.
St Etienne manager Christophe Galtier confirmed earlier this month that Gradel would be allowed to leave the club.
'We have several positions where we have double players in, sometimes even more than that," said Galtier.
"Max-Alain Gradel also needs to play for the upcoming World Cup, so we will not oppose any departure of his."
Having already secured the services of Italian duo Marco Borriello and Antonio Nocerino, West Ham boss Sam Allardyce had previously indicated that he will not be making any more signings during the January transfer window.
However, with the club in the bottom three after winning just one of their last 10 Premier League games, Allardyce may seek to further strengthen his squad.