Meanwhile, the Gunners are supposedly interested in recruiting a new defender of their own, with River Plate teenager Emanuel Mammana apparently high on their list amid interest from north London rivals Tottenham Hotspur.
Allardyce and the Black Cats are keen on Debuchy, but the Mackems gaffer remains unconvinced Arsenal will sell, according to Colin Harvey of the Daily Star: "I am not sure he wants to leave Arsenal or that Arsene Wenger would sell him."
Allardyce has reportedly been making moves to bring Debuchy to the Stadium of Light. He's keen to beat Aston Villa manager Remi Garde, one of Wenger's former players, to the 30-year-old defender.
Despite Allardyce's reservations, Debuchy has been far from shy about his desire to leave the Emirates Stadium. In his article for the Star, Harvey cited Debuchy's recent interview with French publication L'Equipe, during which the full-back made his intentions clear: "Whatever happens, I want to leave, even on loan. I can't stay at Arsenal."
But even though Debuchy lost his starting place to young Catalan Hector Bellerin last season, it still makes sense for Gunners chief Wenger to keep him around, at least for the remainder of the season.
Debuchy is an experienced and capable defender and an excellent deputy for Bellerin. Arsenal lack other natural full-backs who can help, with Kieran Gibbs better suited on the left side.
Calum Chambers has played right-back since joining the Gunners in 2014 but has endured his struggles at the position. His flexibility to play in midfield is also more useful to Arsenal, given the club's ongoing injury problems in the middle.
Santi Cazorla and Jack Wilshere remain injured and are "at least a few weeks away" from returning, according to Wenger's comments on the team's official site. So, Chambers is likely to be called upon to cover in the middle before he plays full-back again.
Debuchy has struggled at times this season, notably in away games against Bayern Munich in the UEFA Champions League and Spurs in the Capital One Cup.
But keeping him around until the end of the season will ensure some experienced cover at the back for Wenger's title-chasing squad.
Arsenal and Spurs Chase Mammana
This isn't the first time the Gunners have been linked with Mammana, with another report in the Telegraph crediting them with interest in the 19-year-old last April.
Wenger's squad could use a promising young defender. Current centre-backs Per Mertesacker and Laurent Koscielny are both ageing (31 and 30 respectively).
Brazilian Gabriel has struggled to unseat either player, but sooner rather than later, Wenger must start thinking about the future of Arsenal's defence.
As a natural central defender, Mammana has the right credentials, after standing out for his country's U20 side last year, according to John Cross of the Daily Mirror. The same report also stated how Mammana already owns a senior cap with Argentina, a sign of his immense potential.
Beating their neighbours to a highly rated young defender would be a real positive for the Gunners, giving them a player hopefully capable of refreshing the back four in the years to come.
Until then, though, Wenger will have to rely on experience at the back, meaning Debuchy should still have a role to play.
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