The Chilean has just one year remaining on his contract at the Emirates Stadium but, despite being keen to leave the club, manager Arsene Wenger has insisted that he is going nowhere.
The Gunners are understood to be willing to lose Sanchez on a free transfer next summer rather than sell him during the current transfer window, but Monaco are reportedly planning to test their resolve with a £45m bid for the 28-year-old.
The Ligue 1 champions could use Arsenal target Thomas Lemar as a makeweight in any deal, with the Gunners having seen two bids for the winger rejected already this summer.
In-demand teenager Kylian Mbappe is also expected to leave in the coming weeks, with Real Madrid interested, so Leonardo Jardim is eager to sign a player to fill the void.
Monaco's sales this summer mean that they would be able to afford a hefty transfer fee for Sanchez, in addition to being able to offer him a £300,000-a-week tax-free salary and Champions League football.