According to MailOnline's Simon Jones, the Blaugrana have offered an initial £55 million along with a further £5 million in add-ons.
Barca are said to be keen to improve their attacking options because they believe "their squad still lacks greater quality in depth and attacking flair".
The Catalan giants won La Liga and the Copa del Rey last year, but they missed out on Antoine Griezmann this summer when he opted to stay with Atletico Madrid.
A fine campaign with the Blues saw Willian return 13 goals and 12 assists in all competitions.
Squawka Football provided the numbers behind the kind of impressive performance he produced during the season:
At the FIFA World Cup, he was deployed on Brazil's right flank, but with Neymar, Philippe Coutinho and Marcelo on the left, he had a fairly quiet tournament for much of the Selecao's time in Russia.
He did impress in their 2-0 win over Mexico, however, and played a key role in a superb team goal:
Goal's Nizaar Kinsella remarked on one of Willian's other contributions in that match:
The winger is a strong, hardworking and reliable player, and he would probably be a good fit under Ernesto Valverde at the Camp Nou, where function and flair can now be found in equal measure.
The Brazilian will be 30 by the time the new season starts, though, and £60 million is an enormous figure to pay for him, even in today's market.
If Chelsea still hold out for a higher fee, Barca should consider a cheaper option who could yield a similar return.