He is yet to agree a new deal with the Spanish champions and as things stand will be available on a free next summer.
Messi has played his whole professional career at the Nou Camp and has always insisted that he is happy to stay with the Catalans until he retires.
Gazetta Dello Sport says Inter are ready to offer Messi a humungous package to try to sign him up.
The report claims the Nerazzurri are willing to hand the 33-year-old £60million a season and would offer him a four year deal.
Just days ago Inter's owners hinted at a summer move for Messi by using his picture to advertise a Serie A game in China.
Both Inter and Chinese streaming service PPTV are owned by the Suning group.