Hammers boss David Moyes is desperate to add some more strength and depth to his midfield department before the transfer window closes and he has set his sights on acquiring the services of Grenier.
It is understood talks over a new contract for the 27-year-old have broken down and, with his current deal expiring at the end of the season, Lyon are willing to let him go to ensure they receive some money.
West Ham, who have reportedly already offered a pre-contract for a summer move, are believed to be able to sign the midfielder for around £1million this month and they are interested in this option.
If France international Grenier were to make the switch to the Premier League, it is understood he will put pen-to-paper on a three-year contract at the London Stadium.
Uwe Rosler was sacked by Fleetwood following a 3-0 reverse at Doncaster which condemned Town to a seventh straight defeat in all competitions.
Cambridge interim head coach Joe Dunne was pleased with how his side adapted to beat Grimsby 3-1.
Southend manager Chris Powell led the applause for Jason Demetriou after the right-back scored twice in the 3-1 win against Portsmouth at Roots Hall.
Morecambe boss Jim Bentley was the happier manager after Gregg Wylde's 90th-minute strike earned his side a point in their 1-1 draw with Forest Green.
Manager Gareth Ainsworth described Wycombe's 2-0 win at Cheltenham as a thoroughly professional performance.
Skipper Richard Wood scored the winner just before half time as in-form Rotherham pulled off a 1-0 victory at League One leaders Shrewsbury.