Talks between Real Madrid and Rodri have stalled in recent days, multiple sources have told ESPN, with Barcelona monitoring the situation as they consider a move if negotiations break down.
ESPN reported last month that Madrid were considering signing the Manchester City star -- who has a year left on his contract at City -- having previously ruled out a bid.
However, Madrid's interest has not developed into an agreement with City, and there is now a degree of pessimism at the club around a deal for Spain's World Cup-winning captain.
Sources close to Madrid now believe Barça are emerging as serious rivals to land the 2024 Ballon d'Or winner.
However, a source close to Barcelona told ESPN that while the Catalan club are keeping an eye on the situation, they have made no formal offer to the player.
Barça's priority remains the signing of a centre-forward, sources said, and they have not given up on a deal for Atletico Madrid's Julián Álvarez.
Rodri is keen to return to Spain this summer, despite City's desire to renew his deal, and the midfielder would be open to the possibility of joining either of the LaLiga giants.
Madrid believe Rodri's signing is now more complicated, given the potential competition from Barcelona, but the club are willing to make one last effort to get a deal back on track.
But a source has said Rodri has concerns that Madrid coach José Mourinho had not been entirely convinced about his signing from the beginning.
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Rodri, 30, underwent minor surgery on a back problem last week, and is currently on holiday after Spain's victory in the World Cup final on July 19.
He excelled at the tournament, picking up the Golden Ball for Best Player, having previously struggled for fitness and form since suffering an ACL injury in September 2024.
A source told ESPN that Barcelona believe the player would fit comfortably into Flick's system, with room in their midfield given the injury suffered by Frenkie de Jong.
Rodri is a fan of Barça's playing style and has good relationships with the dressing room, sources said.
Real Madrid have invested heavily in this summer's transfer market, signing Marc Cucurella, Bernardo Silva, Ibrahima Konaté, Denzel Dumfries and Carlos Espí, while they are close to bringing in Yan Diomande from RB Leipzig.
Barcelona have signed Anthony Gordon and Karim Adeyemi, and have been heavily linked with a move for Álvarez.
Information from ESPN's Alex Kirkland and Rob Dawson contributed to this story.
