The 22-year-old has also been linked with a switch to Manchester United and Chelsea this summer but has reiterated his commitment to Sporting for the foreseeable future.
The Gunners had tabled an improved bid for the Portugal international on Saturday as Arsene Wenger looked to recruit a midfielder with physical presence.
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But the 22-year-old - who has also been linked with Manchester United and Chelsea this summer - has now gone public to reiterate his commitment to Sporting for the foreseeable future.
"I am under contract with Sporting and I'm a Sporting player," he told A Bola. "There is no way I can say I am going to leave. I am fine at Sporting."
Arsene Wenger will spend the final few hours of the transfer window refereeing a charity 'Match For Peace' in Rome and club sources do not expect a busy day in terms of arrivals.