Talks over a deal have therefore halted for now.
It remains to be seen whether Barcelona return with a higher offer but their interest in signing him remains.
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Sources in Spain believe Rodri would prefer a move to Barcelona over Real Madrid.
City do not want to sell Spain's World Cup-winning captain and instead want to agree a new contract with him.
If Rodri asks to leave, however, then they would not stand in his way.
Chelsea's Fernandez would be 'dream' signing for Maresca
As City assess their options for a replacement, Sky in Italy have reported that Chelsea's Enzo Fernandez would be boss Enzo Maresca's "dream" signing.
Sky Sports News reported earlier this summer that Fernandez was not an option they were looking to sign, however, as they have been looking to keep hold of Rodri.
The fluidity of the situation, therefore, means it remains to be seen whether circumstances change with regards to Fernandez, as City may look to bring in two players if Rodri joins Barcelona.
Carra: Don't sell Rodri for £40-50m!
Sky Sports' Jamie Carragher:
"£40-50m? If you're Manchester City, what's the point in selling them? You might as well just keep them for the extra year.
"He's as good as anybody in that position in the Premier League, probably the best. Seeing what he was like in the World Cup, it was like he was back to his best.
"I thought the second half of the season him and Bernardo Silva were fantastic and almost took Manchester City to the title.
"So, if I was the manager of Manchester City, I'd be keeping Rodri at that price. It'd have to be something really special to want me to lose arguably the best midfield player in world football.
"I think it'd be really tough for them to win the Premier League without Rodri, even though they've got a lot of midfielders. He's the best, even though they've brought Elliot Anderson, who is one who I'm sure will learn off Rodri.
"And for me, I'd see those two as a pair. I see Elliot Anderson as almost the replacement for Bernardo Silva, who's gone to Real Madrid. So, I think that'd be a big hole. It'd be really difficult to fill.
"When Liverpool won the title under Arne Slot, Rodri got an injury in that season very early on, which helped Liverpool and was to the detriment of Manchester City. So if I was Manchester City, I'd be keeping him and letting him go on a free.
"City paid £60m for him. If they're getting £40m back now, what can they do with that type of money? Not a lot, really. So, I'd rather keep him."
How much of a loss would Rodri be to Man City?
Sky Sports' Laura Hunter:
Massive. Despite his age, and recent injury record, Rodri remains the most complete central midfielder to have operated in the Premier League over the past five years. "The best midfielder in the world by far," maintains Pep Guardiola.
Under Guardiola, City's in-play strategy revolved entirely around him, always impressing with his use of the ball and ability to receive in tight situations. He set the tempo for his team and controlled games as a result, doing the same for triumphant Spain this summer.
There is no current player who can condition a game like Rodri, and it is no coincidence City failed to retain their Premier League crown in the two seasons he was injury-plagued.
So, how is he replaced?
In short, he isn't.
But in Anderson City have got a surrogate of sorts. The England international might not yet be the dictator Rodri is but he can evolve to become something close. As a box-to-box midfielder, the 23-year-old is putting up numbers that at least mean he is comparable.
