The five-times Ballon d’Or winner will run down his current contract which ends in the summer of 2018, Spanish newspaper Marca reported.
According to the newspaper, the star striker made the decision to move on while on holiday with his family in Ibiza.
Messi reportedly told his father to inform Barcelona President Josep Maria Bartomeu of his decision in July.
Bartomeu is said to have reacted with “understandable shock” and called on the Argentine forward to think again, Marca reported
Messi was slapped with a 21-month suspended sentence for hiding more than £3 million in tax havens.
Messi, 29, joined the Spanish club at age 13.
He broke into the first team aged 17 after rising through the Barcelona academy.
He is now one of the most decorated footballers of all time, with five Champions League and seven La Liga titles in his trophy cabinet.
Fellow striker Neymar signed a five year deal with the club last month.
Bartomeu was confident that Messi would soon follow suit, saying: “Messi? Not yet. But we will have the negotiation with Messi in the next couple of months.”