Cherries boss Eddie Howe is a big fan of the 22-year-old but now Palace have joined the fight for his signature, with the Eagles still also retaining some hope of keeping star man Wilfried Zaha at the club.
It is reported that the Eagles are willing to offer former Bees loanee and fan favourite Sully Kaikai as part of the deal in order to seal the transfer.
Watkins has scored ten goals in 43 appearances for the Bees after signing from Exeter City on a four-year deal at the start of last season.
At the end of the 2016/17 season Watkins received the EFL Young Player of the Year, becoming the first player to receive the honour whilst playing in League 2. The award was most recently won by Fulham's Ryan Sessegnon.
Roy Hodgson is aiming to bolster his attack after losing Bakary Sako on a free and with fellow winger Andros Townsend linked with a move back to Newcastle United, signings need to be made.
Palace have also been linked with a move for Diego Laxalt. The 25-year-old Uruguayan currently plays his football for Genoa and is valued at around £15m, but after solid displays at the World Cup that figure is expected to rise.
Laxalt, who was played at left back for his country this summer, can also play further forward as a winger. Zaha has reportedly rejected a new contract offer at Selhurst Park with the club aware that teams are monitoring his situation.