Arsene Wenger said earlier today that Sanchez will honour his contract with Arsenal and "respect his decision".
Arsenal risk losing Sanchez for free next summer with his contract running out in 2018.
Manchester City have repeatedly been linked with a move for the Chile international this transfer window.
But The Sun claim Monaco are willing to test Arsenal's stance over Sanchez's current situation with a £45m bid.
Arsenal are understood to be strongly interested in signing Thomas Lemar from the Ligue 1 champions.
The Sun report suggests Monaco will use Wenger's interest in Lemar as part of the deal to try and sign Sanchez.
Arsenal have so far only signed Alexandre Lacazette and Sead Kolasinac this summer with the latter joining on for a free.
Monaco meanwhile have lost multiple star players from their side which won the title last year.
Kylian Mbappe also looks likely to be leaving the south of France with reports coming out yesterday saying he has told Monaco he wants to leave.
But Monaco are reportedly ready to make Sanchez their marquee signing of the summer by offering wages of around £300,000-a-week.
Sanchez enjoyed his best season at the Emirates last campaign, scoring 30 goals and providing 19 assists in all competitions.
But Arsenal will be without Champions League football this year for the first time in 19 years.
