The Gunners have keen to bolster their centre-forward options which currently includes Kai Havertz and Gabriel Jesus.
Arsenal have been linked with a number of possible targets, including Juventus' marksman Dusan Vlahovic, who could potentially be available on a cut-price deal in 2025.
The Gunners have also been mentioned as a potential suitor for Gyokeres and were credited with an interest in the summer transfer window.
While they were put off by the player's £86m release clause, they continue to be linked with Sporting's prolific marksman ahead of the January window.
Sporting president refuses to rule out Gyokeres departure
Sporting's president has now commented on the possibility of his star striker leaving in the winter transfer market.
Varandas acknowledged that a departure would be "difficult", but he stopped short of refusing to rule out a January exit for Gyokeres.
"I would say it is very, very, very difficult," Varandas told RTP. "There is nothing impossible in life, but I would say it is very, very, very difficult.
"He has scored many, he continues to score, it would be very, very, very difficult, very difficult."
Why are Arsenal interested in Gyokeres?
While it remains to be seen whether Gyokeres leaves in January, it is clear why he has emerged as a potential target for Arsenal and several other top European clubs.
After joining Sporting from Coventry City in 2023, the Sweden international went on to rack up 43 goals and 15 assists in 50 appearances during his first season with the Portuguese giants.
Gyokeres has carried his impressive form into the new season, having scored 12 goals in as many matches, including 11 strikes in eight league appearances.
The 26-year-old has also showcased his goalscoring talent at international level, netting four goals in Sweden's opening four games of the 2024-25 UEFA Nations League.
Gyokeres will get the chance to impress Arsenal when Sporting take on Mikel Arteta's side in a Champions League clash on November 26.