The 23-year-old spent the 2017-18 campaign on loan with Swansea City in the Premier League, but the Portugal international found it difficult to make his mark at the Liberty Stadium.
Sanches has rediscovered his form at Lille, though, scoring five times and providing five assists in 59 appearances for the reigning French champions.
The midfielder has also made a strong impression for Portugal at Euro 2020, and Liverpool are believed to be considering making an offer after this summer's competition.
However, according to 90min, there is also genuine interest from Arsenal, with the Gunners making contact with Lille to discuss his price tag in the current market.
The report claims that Sanches is open to a return to the Premier League, as he is keen to prove himself in England's top flight after struggling at Swansea.