According to Ed Malyon of The Independent, the La Liga side have made it clear the Argentina international is "not for sale" despite the fact he is reported to have a release clause of €20 million (£17.5 million) in his current contract.
"A high-ranking Sevilla official told The Independent on Monday that Banega is not for sale under any circumstances, and they are planning talks with the player on his return from the World Cup to reassure him that the project at the Sanchez Pizjuan is the best place for his talents," it's added.
Malyon also reported that Sevilla believe the Gunners have "higher priority" transfer targets this summer, whereas Banega is a linchpin in midfield for the Andalusians. Nevertheless, the playmaker is said to be "considering his options."
New Arsenal boss Unai Emery knows the 29-year-old well. Emery was in charge when Banega arrived at the club in 2014, and together they were key to Sevilla winning the UEFA Europa League twice in succession—Emery steered Sevilla to success in the same competition the campaign before, too.
Banega has long been rated as one of the most gifted footballers in La Liga. As we can see below, Phil Kitromilides from the Spanish Football Podcast has been baffled by Argentina's reluctance to use the midfielder at the current FIFA World Cup:
Dermot Corrigan of ESPN FC also commented on the decision, as well as Banega's ability to play incisive passes:
Since returning to La Liga in 2014 after a tough spell at Inter Milan, the Argentina man has been reminding the football world what a talent he is.
While Banega may no longer be as mobile as he once was, his ability to manage a game has got better and better as his career has gone on. He's such a difficult player to take the ball off and the Sevilla man can pull defences apart when given time to pick passes.
Arsenal haven't had a player capable of doing that from an orthodox midfield position since Santi Cazorla; he left the club recently following a prolonged spell on the sidelines due to injury:
With Frances Coquelin leaving the club in January and Jack Wilshere also released recently, Emery will have to bolster his midfield options.
Banega would be a fine start. After five prosperous years at Sevilla, he would surely be intrigued by the prospect of a switch to the Premier League, and it would be fascinating to see if he could implement his elegant style on the frenetic landscape of top-flight English football.
Emery knows how to get the best of the midfielder, something that would surely appeal to Banega if Arsenal were to trigger his buyout clause. Sevilla may be keen to keep hold of him, but they're a club that have traditionally cashed in on even their most important players for the right amount.