The Blues have been linked with a host of managers, with Pep Guardiola the No 1 target for owner Abramovich.
The Russian chief is also said to be considering Jorge Sampaoli - who led Chile to the Copa America title last year.
The 55-year-old has managed Sanchez with Chile since 2012 and is reportedly ready to shun other offers to be reunited with him in west London.
But Arsenal would be loathe to sell their star man to their rivals, while there is no guarantee Sanchez would even want to leave for the Blues.
Sampaoli is desperate to quit as Chile boss and has accused the country's new ANFP president Arturo Salah - who took over after former chief Sergio Jadue resigned amid a corruption investigation - of treating him like a hostage.
"In this environment I don't want to work, or even live, in the country anymore," he said.
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