Willian, 31, had been expected to announce whether he is leaving the club this week but sources have told ESPN he will now wait until after Saturday's Champions League round-of-16 second leg at Bayern Munich before deciding his future.
Blues head coach Frank Lampard wants Willian to stay and has been pushing the club to find a breakthrough in negotiations which have been deadlocked for months.
Willian is happy in London and willing to sign a new deal but Chelsea have so far only offered him a two-year extension. Sources have told ESPN that the Brazilian is adamant he will not accept less and will depart in search of a new club if Chelsea are unwilling to match his demands.
Arsenal are monitoring the situation closely and have indicated they are ready to discuss terms should Willian opt to move. The player's agent, Kia Joorabchian, claimed last week Willian has five clubs interested in signing him including two in the Premier League.
One of those is thought to be the Gunners, with Willian's experience and professionalism viewed as important assets. Arsenal's FA Cup final victory over Chelsea last weekend -- which Willian missed due to an Achilles injury -- means they can at least offer Europa League football next season.
There are at least two other European clubs interested while David Beckham's Inter Miami are also interested in luring Willian to the United States.