The west London club still have €150m left to spend on a player whose contract at the Santiago Bernabeu runs until the end of June 2020.
Spanish outlet Sport claim the Blues are now most interested in the Colombian and are prepared to spend a whopping £78m on the 25-year-old.
Rodriguez has fallen down the pecking order at the Bernabeu since the arrival of Zinedine Zidane and is interested in leaving the La Liga side.
Conte is keen to add more new faces to his Blues side before the transfer window slams shut on August 31.
The Italian has so far brought N’Golo Kante and Michy Batshuayi to Stamford Bridge but he is keen to bring even more creativity into his side.
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