FotbollDirekt, which refers to several English sources, states there is no interest from Arsenal.
Edu, Arsenal's sporting director, is looking for a different type of striker this summer.
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Gykeres has a buyout clause worth 100m.
The striker's contract expires in the summer of 2028 - but Sporting are said to be trying to extend it and at the same time raise the buyout clause.
Gyokeres has also been linked with Manchester United, Liverpool, Milan, Atltico Madrid and Paris Saint-Germain.