The Brazilian is entering the final year of his contract at Stamford Bridge after six years at the club.
Sky Sports News reports on Monday that both La Liga giants have seen offers of £35 million turned down by the Premier League club.
The Blues are said to be determined to retain the 30-year-old despite the risk of losing him on a free transfer next summer.
They hope to tie down the Brazil international on a new two-year deal and ward off interest from Barca, who have long been admirers of the forward.
The Spanish champions had three bids for Willian turned down last summer before launching another failed attempt to secure his signature in January.
Chelsea are currently unable to bring in new players as they serve the first part of their FIFA-imposed two-window transfer ban.
Willian made 32 Premier League appearances for the club in the 2018/19 season, scoring three goals and providing seven assists.
Read more...
QUIZ! Can you name the 25 most valuable players in the world aged 21 and under?
FEATURE The legacy of USA '94: why two big misses didn't prevent a World Cup hit
