Gueye has impressed in Ligue 1 this season, with his powerhouse performances catching the eye.
He is now said to be of interest to a number of clubs across Europe, and Palace and West Ham are reportedly among the suitors.
Lille are apparently demanding £7 million for the central-midfielder, but neither of the Premier League duo are put off by the French side's asking price as they feel he could be pivotal to their chances next season.
Gueye is regarded as one of the finest talents in the French top-flight, after his excellent displays protecting the Lille defence this term.
If either West Ham or Palace were able to land the Senegalese international, it would be a good start to an important summer for both clubs.
The London pair are both aiming to make big strides next season, as they look to compete for a European spot, and Gueye would add undoubted quality to their respective squads.