The Stamford Bridge club have made their move after London rivals Tottenham held talks with Willian earlier on Thursday.
Spurs were ready to splash £30million on the midfielder, who has been offered a five-year contract at White Hart Lane.
But the 25-year-old, who was also linked with Liverpool earlier in the summer, appears to be keeping his options open.
Should Willian move to Stamford Bridge, it will raise further question marks over the future of Juan Mata, who has been linked with a move to Manchester United as a makeweight in any possible deal for Wayne Rooney.
Chelsea have made Rooney their top striker target this summer and have already have two bids rejected. Although United insist they won't sell Rooney to a direct Premier League rival, they could be willing to do business should Mata form part of the deal.
United have so far drawn a blank in their efforts to sign a creative central midfielder, having had two bids for Cesc Fabregas rejected.
And should Willian join the Blues it will represent a real blow to Spurs boss Andre Villas-Boas, who had been hoping to sign the player to offset the likely loss of Gareth Bale to Real Madrid.