Despite the Chilean having just over a year left on his contract, Arsenal manager Arsene Wenger has stressed that their star man will not be sold to any of their Premier League rivals.
However, according to The Independent, it appears likely that the Blues will launch a bid to try to tempt the North London outfit to cash in on a player who can currently depart the club for free next summer.
It is unclear what fee, if any, would be regarded as high enough to test the resolve of Arsenal, who may decide to take their chances of trying to agree a new deal with the 28-year-old at a later date.
It has been claimed that the player prefers to remain in London, suggesting that staying at Arsenal or making a controversial switch to Chelsea - even next summer - are his main two options.
Sanchez has featured in all 37 of Arsenal's Premier League fixtures this season, contributing 23 goals and 10 assists.