Having failed to make an impact at Stamford Bridge since his £27m move from Fiorentina in January, the Colombian international has been linked with a return to Italy with both Roma and Inter Milan said to be interested.
However, reports in Spanish newspaper Sport claim that Cuadrado may find himself heading to Catalonia after the Blues got in touch with the La Liga champions about a possible transfer.
Barcelona are still under a transfer embargo until January 2016, and though they cannot sign any players in the summer, they are allowed to strike deals to go ahead in the new year.
Blues boss Jose Mourinho wants to get Cuadrado off the books at the first opportunity, and has identified Barca as prospective buyers.
He is expected to head up an exit list that includes the likes of John Obi Mikel, Filipe Luis, Loic Remy, Oscar and Mohamed Salah, who have all fallen down the pecking order – or failed to make it on to pecking order, as in Felipe Luis’s case – at some point over the past 12 months.