The playmaker’s future at Real Madrid remains uncertain following a difficult spell under Zinedine Zidane.
Rodriguez has struggled for game time and Madrid chief Florentino Perez is ready to offload the 25-year-old.
The La Liga giants want just over £50million, and according to Spanish outlet Diario Gol, City are in negotiations.
Pep Guardiola is said to be pursuing a deal at the Etihad.
Pep Guardiola is looking at transfer targets for next season at Manchester City
Manchester City are only offering £34m, and Perez is not willing to lower his asking price
And the ex-Barcelona boss believes he can get the best out of Rodriguez in the Premier League.
Following City’s Champions League exit to Monaco and fall behind Chelsea in the title race, Rodriguez is being eyed as a solution for the team next term.
However, Manchester City are only offering £34m, and Perez is not willing to lower his asking price.
Another issue is the mass competition faced for his signature, despite Guardiola’s supposed pact.
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Manchester United and Chelsea have apparently asked about Rodriguez’s situation at Madrid.
But similarly to City, Jose Mourinho and Antonio Conte are failing to reach an agreement over the Colombia star’s fee.
Inter Milan, Juventus, AC Milan, Bayern Munich and Paris Saint-Germain are also interested.
For now, City will not yield, but Rodriguez could still head for the Manchester club this summer.