Barcelona have repeatedly urged patience from within the club in recent days, convinced the transfer would eventually be completed despite having to raise their offer twice already.
An initial €45m proposal was increased by 25% to €60m before this latest improvement takes the total to around €70m, with club executives hopeful City's Ferran Soriano and Hugo Viana, who are leading negotiations for the Spain captain, will now find the offer acceptable.
Rodri, who lifted the World Cup with Spain on July 19 at the MetLife Stadium in New Jersey, made known his desire to leave City months ago. Barcelona are preparing a four-year contract for the Madrid-born midfielder, who would return home seven years after leaving for the Premier League having won almost every honour available to him, including European Championship and World Cup MVP awards across the past two tournaments — a run widely credited with anchoring Spanish football's recent golden generation.
Rodri travelled to Manchester via Liverpool on Thursday night and reported for pre-season training on Friday morning as scheduled.
New Manchester City manager Enzo Maresca greeted him warmly but could not confirm whether the midfielder would remain at the club.
"I gave him a big hug," Maresca said. With Barcelona's third offer now on the table, the deal appears closer than at any previous point this summer, contingent only on City finally accepting the improved terms.
Rodri's potential departure would leave a significant void in the middle of Maresca's midfield, though City are simultaneously closing in on a separate deal for Lille's Ayyoub Bouaddi that would offer some compensation.
Barcelona, for their part, view Rodri as a marquee addition capable of dominating La Liga in the way he has dominated the Premier League since his move from Atlético Madrid in 2019, and are confident that this latest, improved offer represents their best and final attempt to get a deal over the line before the transfer window closes on September 1.