Sanchez has refused to sign a new deal at the Emirates despite the club opening talks last year.
There is a £180,000-a-week deal on the table but Sanchez wants parity with the league's top earners and is looking for a package in the region of £250,000-a-week.
He was dropped from the Arsenal team on Saturday but came on at half time in the 3-1 defeat to Liverpool.
Alexis Sanchez is wanted by Chelsea as his contract stand-off with Arsenal continues
It emerged afterwards that Sanchez had been dropped for his behaviour in recent weeks, believed to include rowing with team-mates over their form.
However Arsene Wenger denied those suggestions on Monday and insists his future is at the club.
But the Daily Mirror report that Chelsea will test the waters with a summer bid if Sanchez is no closer to signing an extension in north London.
Alexis Sanchez has started Arsenal's last two matches on the bench
There is also interest from Europe with PSG, Juventus and Atletico Madrid keen on the 28-year-old while Manchester City are also monitoring his situation.
Sanchez has scored 17 Premier League goals this term but his form has dipped in recent weeks as Arsenals title charge has come to an end.
He was visibly frustrated during the 5-1 humiliation against Bayern Munich in the Champions League but is likely to return to the starting XI tonight for the second leg.
Alexis Sanchez spotted in training after Arsenal bust-up
Mon, March 6, 2017
Alexis Sanchez has been spotted in training after his reported bust-up last week
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Alexis Sanchez has been snapped in training after his bust-up last week
And Wenger has called on him to show 'lucid rage' as Arsenal bid to overcome a four-goal deficit.
“What you expect is that you go out there with lucid rage," Wenger said.
“Total commitment, but not a silly one because in our game you always need to make intelligent decisions, to make the optimal decision and for that you need a good combination of commitment but as well I think to keep your vision.”