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But Walcott’s former club Saints and the other three Premier League outfits only want to take the 28-year-old on loan.
Walcott’s wages of around £110,000-a-week are a major stumbling block to him landing a permanent move.
Once an England regular, Walcott has not featured for his country since coming on as a sub in the 2-2 friendly draw in Spain in November 2016.
He has also fallen way down the pecking order at the Gunners and it is understood he has accepted he needs to move away from the club he joined for £5m some 12 years ago.
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Wenger has shown before he is willing to let an England international go out on loan, allowing Jack Wilshere to go to Bournemouth last season.
Arsenal are also lining up a £2.2m bd for 20-year-old Greek defender Konstantinos Mavropanos of PAS Giannia.
He is not seen as an immediate first-team addition and could even be loaned straight out to Werder Bremen.