Conte's future at Stamford Bridge has come under increasing scrutiny in recent weeks due to a string of questionable results, including emphatic losses to Bournemouth and Watford in the Premier League and an EFL Cup exit to London rivals.
Reports in recent days have suggested that Blues owner Roman Abramovich has already started the process of finding Conte's replacement and has earmarked former Barcelona boss Enrique as his prime target.
According to top Spanish journalist Eduardo Inda, Enrique has begun thinking about what he might do at Stamford Bridge and has prioritised a raid on his former side in order to bring Suarez back to the Premier League.
The 31-year-old moved to Camp Nou from Liverpool in a £65m deal in July 2014 and went on to form a successful partnership with Lionel Messi and Neymar, which ended when the Brazilian departed for Paris Saint-Germain last summer.
Suarez committed to a new deal with Barca in December 2016 keeping him at the club until 2021, with an associated release clause of €200m (£177.6m).