The 29-year-old joined Barca as a youth player in 2001 and has gone on to establish himself as one of the best footballers of all time, becoming the club's all-time leading goalscorer with 507 goals in 583 appearances.
It is said Messi will sign a three-year contract and will be given the option of extending it by a further year.
Barcelona have been in negotiations with Messi's father for months but last month AS added more uncertainty to the saga after reporting that the five-time Ballon d'Or victor had rejected the first offer made by the club, worth between €30m (£26m) and €35m.
"It's what I always dreamed and what I always wanted".
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Messi is now in Australia, where he is expected to play a leading role as Argentina take on arch-rivals Brazil in a friendly in Melbourne on Friday evening (June 9). Throughout all competitions, the worldwide star netted 54 goals, but it still was not enough to help them capture La Liga or Champions League glory.
Messi staying would obviously be a huge boost for new Barca manager Ernesto Valverde, who must build a side capable of taking La Liga's title and the UEFA Champions League off bitter rivals Real Madrid in 2018. This year it wasn't to be.
Enrique announced in March that he would leave the club in the summer after three years in charge and his final match saw Barca capture the Copa del Rey trophy.