The La Liga leaders have tried multiple times to sign the Brazilian, but have fallen short each time.
Willian is expected to walk away from Stamford Bridge on a free transfer this summer. The 30-year-old's contract will expire in a matter of months, and admits he isn't near to extending it.
Sport says Barcelona know they must sign him now or risk missing out on him before the end of his career.
Willian said earlier this week: "There were no more talks or negotiations. Chelsea said that three years would be impossible, so at the moment it seems unlikely that I'll extend, but nothing is impossible.
"We don't know what might happen. We might suddenly reach an agreement, but what I can say today is that my contract is up and I'll be free to negotiate with any club."