Bristol City have also been keeping an eye on the 25-year-old, who allegedly features on the list of candidates Antonio Conte is considering to replace Asmir Begovic as their number two next season.
However, Bettinelli is unlikely to make the switch to Fulham's neighbours as a result of first team football being limited, something he desperately wants next season having spent most of this year on the bench.
Speaking to GetWestLondon, Bettinelli said: "Its been a long season for me, it hasn't been a great season for me if we're honest.
"I love this club and I've been at this club for a long, long time but I'm going to be turning 25 in the summer and I want to be playing regular football, and this season I haven't done that.
"I've got two more years left and I want to stay here, I love this club and want to play for this club but at the end of the day I need games and if I'm not going to get them that's it, but hopefully I can get them and get this number one shirt."
Chelsea made enquiries last season but Bettinelli rejected a move and it's likely his position on the matter won't have changed this time around.