The Portugal midfielder has asked to leave the Etihad this summer.
The Sun says Bernardo, 26, is keen to play in LaLiga.
And his agent Jorge Mendes is trying to negotiate a move to Barcelona. City currently are refusing to listen to any offers below £45m.
That figure could price out Barca considering their financial difficulties.
But with the departure of economically-draining Lionel Messi, the Catalans now have some funds to at least make a bid.
Bernardo has told friends he will plead with Guardiola to let him go.
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