The Daily Star suggests the Blues have moved to the front of the queue despite interest from a number of Premier League rivals.
Arsenal were thought to want him to team up with international team-mate Alexis Sanchez and compatriot Manuel Pellegrini was also keen to bring the Copa America star to Manchester City.
The 26-year-old made a huge impact at the tournament, playing a key role as they lifted the trophy.
Although interest is high, his club are believed to be reluctant to open talks until after their Copa Libertadores semi-final with Tigres.
However, Chelsea have a good relationship with the club after signing Oscar in 2012 and a fee of around £8million might convince them to do business.
After completing the signing they are expected to turn their attention to further transfer targets.
A £21.5m bid for Barcelona winger Pedro is believed to be on the horizon, while the Blues might return to Spain with an offer for Real Madrid star Isco.
Asmir Begovic, meanwhile, is close to completing his £8m switch from Stoke.