Manager Sam Allardyce is keen to add to his strike force having seen a move for Estudiantes forward Duvan Zapata fall through last week.
Co-chariman David Sullivan expressed the club's desire to add to their attacking options over his son's Twitter account.
"We are happy with the rest of the team - all we need now is pace and goals," the 64-year-old said.
The 26-year-old, who scored six goals in 14 appearances including a debut goal at Upton Park, has also attracted interest from Tottenham and Newcastle.
The French international rejected the north-east club in favour of accepting a £75,000-a-week deal at Loftus Road in the winter transfer window.
West Ham came close to signing Remy in 2010 after agreeing an £11.5m fee with his then club Nice, before a move fell through with the striker opting to join Marseille.