The Argentine's current contract expires in June, which means that the 32-year-old can talk to European clubs about a move away from the Etihad Stadium.
The Sun reports that the Catalan club have been alerted that Aguero is yet to negotiate a new deal with City and are hopeful of beating Spanish rivals Atletico Madrid to sign the striker.
French champions Paris Saint-Germain are also interested in the City man who are now managed by compatriot Mauricio Pochettino.
Aguero, who has spent nine years with City, has won four Premier League titles, five EFL Cups and one FA Cup during a trophy-filled spell in Manchester.
The forward, however, has suffered with a knee injury for the majority of the 2020-21 campaign and has only featured nine times in all competitions so far, scoring just twice.
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