Van Aanholt is one of a several Palace players whose deals run out at Selhurst Park at the end of this season.
Though the club hope the popular Dutchman stays, there is strong interest in Alonso as an alternative with Marseille’s Jordan Amavi also being tracked.
Alonso, 29, has fallen out of favour at Chelsea this season and hasn’t featured since being substituted at half-time when they were 3-0 down to West Brom in September.
There were reports the disgruntled Spaniard then tried to watch the second half of the Hawthorns match on the Chelsea team bus – prompting a furious reaction from manager Frank Lampard.
Palace are hoping to capitalise on a good start to the season under Roy Hodgson by holding talks with a number of players over their long-term futures.
Midfielder Jairo Riedewald is likely to be the first to re-sign this week with discussions also active with Scott Dann, Vicente Guaita, Gary Cahill, Tyrick Mitchell and Van Aanholt.