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Published 22:03 28 Jun 2016 BST
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Having left PSG, he's available and also has had a taste of international management with France between 2010 and 2012.
Keane has appeared at his most relaxed in years and yet has been entirely capable of letting off verbal bombs to keep the Irish players in check.
He has helped Martin O'Neill bring about a culture change, introduce new players and has taken Ireland further than they have achieved in tournament football since 2002. He is respected by a host of modern players, who grew up watching him play, and, most importantly, would take no shit from primadonnas.
He's a shrewd tactician and his charisma would make him a popular choice.
Being West Ham manager though is a stumbling block. It's unlikely he'd leave his position and even if he did, he wouldn't be able to bring Dimitri Payet with him.
His time at Arsenal is coming to an end, too. If Piers Morgan and the anti-Wenger, banner-wielding Arsenal supporters get their way, we could be looking at the last 12 months of his time with the club. Perhaps the England job would tempt him.
And don't forget, the winners of that 2014 World Cup were his legacy. Joachim Low inherited the team having worked alongside him. If people can get over the fact that he's German, he might be a sensible choice.
The one thing England really need right now is a system that they stick to and, as he showed last season, Ranieri certainly knows how to get a team playing in a certain way.
Would he leave Leicester? Maybe. The club's owners and fans wouldn't want it, but given his miraculous achievement last season, perhaps they'd reluctantly accept it.

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