Sanchez has been a revelation for Arsenal since joining from Barcelona for £35m in July 2014.
And after a slow start to the season, the Chilean forward has hit form with six goals in his last three games for the Gunners.
But Spanish magazine Don Balon claim Real want to lure Sanchez back to La Liga with a £37.3m bid - and could also offer winger Denis Cheryshev to sweeten the deal.
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