Barcelona are preparing to submit a third offer worth about £60m for Rodri, but that bid is still expected to be £10m short of Manchester City's valuation of the Ballon d'Or winner.
The transfer could also affect the pace of any move for Chelsea's Enzo Fernandez, with Enzo Maresca pressing to bring in the midfielder.
City are open to allowing a long-serving club great such as Rodri to realise a personal ambition of playing in Spain again following this summer's World Cup triumph.
Rodri with the Golden Ball award at the World Cup.
However, City, who are planning changes to their midfield, have so far been left both confused and frustrated by negotiations with both Barcelona and Real Madrid.
Neither club has come close to the £70m asking price, with Real Madrid's hesitation eventually leading Rodri to choose Camp Nou over the Bernabeu.
Even so, the same problems have surfaced in that move too, with Barcelona having already made two offers well under what City are seeking, one of them described as "derisory".
Those discussions are taking place alongside talks over Fernandez, for whom Chelsea are demanding £120m.
Although the club had set a 5pm deadline on Friday for interest in the midfielder, after which they would keep him, the current feeling is that enough progress has been made in the talks.
There is also a recognition that City will not move fully forward until Rodri's future is resolved, which is likely to stretch beyond Chelsea's deadline.
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