They are also keen to add Ilkay Gundogan and Ruben Neves to their midfield ahead of next season.
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With Messi set to leave Paris Saint-Germain in the summer, a return to Barcelona is very much a possibility.
Previous speculation has linked him with a comeback at Camp Nou, and this summer's transfer window could seal a deal completed.
As reported by SPORT, Barcelona are offering him around £22million-per-season on a two-year contract, although he could leave in 2024.
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Messi scored 672 goals in 778 appearances across all competitions during his previous stint at the Catalan side.
His contract at PSG will expire at the end of the season, with Gundogan also set to have the same fate at Manchester City.
The German international is said to be searching for a new adventure away from the Premier League, and could sign for Barcelona for two seasons.
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Gundogan is one of the Premier League's top players, and the City man may not be the only midfielder to switch England for Spain.
Wolves star Neves is another target for Barcelona, who tried to sign him last summer but were unable to do so due to his current club's demands.
However, it is reported that such a transfer may be easier to finalise this time round as his fee could be less.
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Neves' agent Jorge Mendes also has a 'good relationship' with the Barcelona board.
The summer transfer window could therefore be one of reuniting and change for the La Liga leaders.
Barcelona are just three points away from winning the league once again, and look set to repeat their achievements next season with bold acquisitions.