The Sweden international has been superb for Sporting this season, having arrived last summer from Coventry City.
A Bola reports Arsenal and PSG are now intensifying their pursuit of Gyokeres.
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There are already several Premier League clubs who regret not trying for Gykeres last year before his departure for Lisbon.
The 25-year-old's buyout clause is set at 100m.
Gykeres has a contract with Sporting CP that runs to 2028.