The European champions are ready to put together a £40m package in an effort to lure the Uruguayan striker to Stamford Bridge, but the report claims Manchester City are also interested in the Liverpool star.
Suarez has been in sparkling form for the Anfield club this season, and leads the Premier League with 10 goals already this term.
But the 25-year-old has proven to be a controversial figure since completing his £22.8m move from Ajax in January 2011. Suarez received an eight-game ban f racially abusing Patrice Evra last season, and has been in the spotlight recently for his alleged propensity to dive in an attempt to win penalties.
Suarez was signed in the same transfer window as Andy Caroll, but the Uruguay international has proved to be a huge success at Anfield compared to his former-teammate.
As a result, Chelsea are thought to be considering a move for the star striker, as £50m forward Fernando Torres continues to struggle.
The Spaniard had another difficult night against Manchester City at the weekend, and despite the arrival of former-Liverpool boss Rafael Benitez, the Blues have been linked with a number of potential replacements.
But Liverpool boss Brendan Rodgers is desperate to keep hold of Suarez, who has been one of the few bright spots for the Reds this season.
Chelsea, like Liverpool, are short of options up front, with just Fernando Torres and Danny Sturridge available to Rafa Benitez as senior striker options.