The World Cup winner has yet to put pen to paper over a new contract at Paris Saint-Germain, where his current deal will expire in the summer.
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The French giants had felt confident that they could tie the Argentine down to a new deal but those chances appear to be slipping away.
Barcelona has emerged as a possible destination in a grand return to the club where he made his name before leaving in 2021 as they could not afford his wages.
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And Barca boss Xavi has made it abundantly clear that he wants to see his old teammate return to the Camp Nou this summer.
Xavi said: "I have already said that this is his home and his doors are open. He is a friend and we are in permanent contact.
"It will depend a lot on him, what he wants to do in the future, what fits in at the club. This is his home.
"The best footballer in the world and the story would always fit."
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Messi's father has already met with Barcelona chiefs this week over a potential transfer back, according to Catalunya Radio.
It is further stated that talks were positive while Barca president Joan Laporta would love to see the 35-year-old come back to the club.
In December, Laporta said about a possible Messi return: "For us he is the best player of all time.
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"We have had him here and I am convinced that in his heart he is a Cule and will always be linked to Barca.
"Whether or not he comes back as a player, at the moment he is at PSG and we would very much love him to come back one day, but we'll see."
Rumours that he might leave PSG for Barcelona this summer were fuelled when his wife Antonella Roccuzzo uploaded a picture of the couple with friends, including Barca pair Jordi Alba and Sergio Busquets, in the city centre.
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According to Marca, the seven-time Ballon d'Or winner had plans to sign a new contract at the Parc des Princes.
However, after winning the World Cup, he has had a change of heart and is considering a U-turn as he is less concerned with winning domestic titles now.
Messi scored 672 times across 778 appearances for Barca. During his time with the club he won ten La Liga titles and four UEFA Champions Leagues.