The freed Leicester City man is the latest right-back to be linked with Crystal Palace, alongside Hamari Traore of Rennes.
Simpson was released by Leicester at the end of last season and can therefore sign for another club, including Crystal Palace, after Thursday's transfer deadline.
The 32-year-old is one of several in his position being linked with moves to Palace, including Tottenham Hotspur's Kyle Walker-Peters and Rennes' Hamari Traore.
But should no move materialise within the next 24 hours then Simpson could be the man to whom the Eagles boss, Roy Hodgson, turns.
Crystal Palace are looking to replace Aaron Wan-Bissaka, who joined Simpson's first club, Manchester United, in a reported £50 million deal earlier this summer.
The Mancunian has also played Premier League football for Newcastle United, Queens Park Rangers and Leicester, with whom he won the title in 2016.
Would the freed Leicester man be a good addition for Crystal Palace?
Have something to tell us about this article?