The 25-year-old came through the youth ranks at Lyon before joining Nice in 2014 for £500,000.
Last season was his most prolific to date, netting 21 goals in all competitions and assisting a further seven.
His form has caught the attention of some Premier League clubs with both Tottenham and Newcastle believed to be interested in the Frenchman.
But according to Sky Sports, the Hammers have now put in a bid for the Ligue 1 goalscorer of £27m.
It also said that one other club has already made an offer of £25m.
Meanwhile, Pellegrini is looking forward to an ‘exciting time at West Ham’ after putting together his backroom team for next season.
The Hammers boss will be assisted by Ruben Cousillas and coach Enzo Maresca, while former Real Madrid and Liverpool goalkeeper coach Xavi Valero will be employed in the same position at the London Stadium.
Fitness coach Jose Cabello will be reunited with Pellegrini following a spell under the Chilean at Malaga, Manchester City and Hebei China Fortune, with Cabello assisted by Felix Cao, who was also on Pellegrini’s staff in China.
“My backroom staff are hugely important to me, as they have been throughout my managerial career,” Pellegrini told the club’s official website.
“I know they will play a pivotal role in helping the team prepare and perform to the best possible level.
“As a group, and as individuals, they bring with them a lot of experience and know-how as well as a great understanding of how we want to do things.
“These are all important factors in putting a successful team on the pitch, which is what we are all striving for.
“I’m delighted we are going to be working together again as we enter an exciting time at West Ham.”
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