Real Madrid had long looked the likelier destination for the 30-year-old, who has just a year left on his Manchester City contract, but Los Blancos failed to move quickly enough and Barcelona stepped into the picture and persuaded the World Cup-winning midfielder to commit his future to the Camp Nou instead.
Agent Pablo Barquero confirmed just over a week ago that Rodri had made up his mind in Barcelona's favour, but the Catalan club have yet to reach agreement with City over a fee.
According to Sky Sports, City have turned down Barcelona's latest offer of around £55 million, with the Premier League champions holding out for closer to £60 million to sanction the sale.
City's stated intention throughout has been to extend Rodri's contract rather than sell, but the club say they will not stand in his way given his clear desire to move, and negotiations between the two clubs continue with a degree of optimism that an agreement can still be reached.
Rodri was named the best player of this summer's World Cup and remains a live contender to win the Ballon d'Or for a second time, underlining the calibre of player City risk losing for a fee some way below what many would consider his true market value.
Since his club-record move from Atlético Madrid, Rodri has scored 28 goals and provided 32 assists in 298 competitive appearances for City, establishing himself as one of the most complete midfielders in world football and a central figure in the club's recent trophy-laden era under Pep Guardiola.
With both parties reportedly hopeful of closing the remaining gap, the coming days are expected to prove decisive in determining whether Rodri completes what would be one of the most significant transfers of the summer.
City are already braced for the possibility of needing a replacement in defensive midfield should the move go through, having spent heavily elsewhere in the squad this window under new head coach Enzo Maresca, who has voiced hope that Rodri could yet remain for the season regardless of how talks with Barcelona ultimately conclude.