According to a report in The Daily Mail, the Blues made a £20.6 million bid for the winger on Wednesday, but it was snubbed by the Italian side as it fell short of his £26.8m release clause.
Cuadrado garnered much interest in the summer following an excellent campaign last season and a solid showing at the World Cup.
In an interview with Goal in September, he admitted holding talks with Manchester United and Barcelona, but a transfer ultimately failed to materialise.
Now, with Andre Schurrle and Mohamed Salah linked with Stamford Bridge exits as reported by The Daily Star, Chelsea manager Jose Mourinho may feel the need to bolster his attacking ranks before the transfer window shuts.
Colombian international Cuadrado represents a sound option, and his current manager Montella seems resigned to losing him.
"I believe it's a simple situation, he has an exit clause that he agreed to with the club: if someone pays it and he wants to leave, we can't do anything about it. But he's off the market for the club and I, but when there is a clause there is nothing you can do,” he was quoted as saying by Sky Sport Italia.
The 26-year-old has been theViola's most consistent performer this season, scoring four goals and adding a further four assists in 16 Serie A starts.
Capable of playing on either wing and possessing fabulous dribbling skills, Cuadrado would certainly offer the West London outfit a new dynamic in the final third and perhaps, most importantly, consistency.