The 27-year-old midfielder, who can play in the centre and the right of the park, has struggled this season with the Trotters, making just 15 appearances, and scoring two goals and registering one assist.
Davies, who has also played for Wolves and Leicester during his career, has made 163 appearances in total for Bolton, but following the appointment of Neil Lennon as manager, it was apparent that he was not in favour at the Macron Stadium.
He appeared in the final four games of the campaign, but spent a 24-game spell prior to that out of matchday squads - with another knee problem doing him few favours.
His potential transfer availability this summer has made Palace well aware of his talents on the market, and the report indicates that Alan Pardew is keen on bringing him to Selhurst Park.
The Eagles are willing to sell Mile Jedinak this summer, and Davies is likely to be brought to the club as midfield cover for first-choice pairing James McArthur and Joe Ledley. He has a wealth of experience of English football, and has proven Premier League ability during spells in the top flight.
Whilst not a marquee signing or one to bring much excitement to Palace fans, making a move for Davies would certainly be a logical move for the South London club.