The Hammers are focused on finalizing the integration of Julen Lopetegui and his staff to replace the departed David Moyes.
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Southampton, where Downes spent a year on loan, want to bring him back permanently.
As the Saints are now in the Premier League, they will be a serious option for Downes.
However, West Ham will have the final say on his future, with Downes stating recently: "I can't say too much, we'll see what happens."