The 25-year-old striker netted 12 goals last season to help the French club win their relegation battle in Ligue 1.
West Ham have reportedly had two bids knocked back for the Denmark international - whose contract expires next summer - the latest of which was £3 million, say The Mail.
But the same publication has now claimed that the French outfit are looking for offers in the region of £4 million for their talisman, whilst crediting Sunderland and Crystal Palace with an interest.
The report goes on to state that Braithwaite has flown to England for extra training sessions with striker coach Allan Russell, who works with Newcastle United's Aleksandar Mitrovic and West Brom's Saido Berahino.
The Hammers need a striker, but fans' appetite has been whetted by names such as Alexandre Lacazette in recent weeks and there would an underwhelming sense if Braithwaite is the only goalscorer to join the Irons this summer.
Big things are happening in and around the East London club and supporters will expect a shift in transfer policy as a result.