Barcelona will return with a renewed offer for Spain international Rodri as negotiations with Manchester City continue, a source told ESPN.
Barça have agreed personal terms with Rodri, who has opted for the Spanish champions over Real Madrid, but have so far failed with two bids for the midfielder.
Sources told ESPN that City value the 30-year-old at more than €70 million (£60m / $81m), with the total package of Barça's latest offer expected to be in that region.
Rodri flew back to England this week after time off following Spain's World Cup victory in North America last month.
Rodri has been the subject of two failed bids by Barcelona so far. East Rutherford, New Jersey. (Photo by Maja Hitij - FIFA/FIFA via Getty Images He met with Enzo Maresca for the first time on Friday, with the new City coach coy on whether the two will get the chance to work together this season.
"As I already said, when the transfer window is open, we can buy players, we can sell players, anything can happen," Maresca said in a news conference ahead of the FA Community Shield against Arsenal on Sunday when asked about Rodri.
"I just saw Rodri now, I gave him a big hug, he looks good. So let's see what happens."
Asked if he will chat to Rodri, he added: "Now we have time to see each other a bit more."
With Bernardo Silva joining Real Madrid on a free transfer and sources telling ESPN City have agreed a deal with Al Qadsiah for Tijjani Reijnders, the Premier League side are keen to strengthen in midfield before potentially losing Rodri.
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They have already signed England's Elliot Anderson from Nottingham Forest for £116 million ($154m), but remain in the market for further reinforcements.
Chelsea's Enzo Fernández and Lille's Ayyoub Bouaddi are both targets, ESPN have previously revealed.
However, Maresca revealed City are unlikely to meet Chelsea's Friday deadline for Fernández, with the Blues demanding £120 million ($161.8m) for the Argentine.