The outgoing Bayern Munich boss has unfinished business with English football having been sacked by Chelsea in September 2022.
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West Ham’s current boss David Moyes is out of contract at the end of this season and recent results have placed him under extra pressure.
Some fans even want him out following a run of eight games without a win in which they have conceded 16 goals.
Hammers’ chiefs are exploring possible options for a change of coach while remaining firmly behind Moyes for the immediate future.
Tuchel will become a free agent in the summer after it was announced he will be leaving German champions Bayern at the end of this season.
But sources close to the German say Tuchel will only entertain ideas to manage at the biggest clubs in England.
Liverpool are losing head coach Jurgen Klopp this summer and with new people on the board at Manchester United, manager Erik ten Hag is facing an uncertain future.
Newcastle United boss Eddie Howe is also under pressure with his team eighth in the Premier League despite the backing of Saudi billions.
Tuchel managed Chelsea for a year and a half from January 2021 and won the Champions League and Club World Cup before helping the club through the most turbulent period in its history.
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Russia’s invasion of Ukraine forced then club owner Roman Abramovich to sell the club amid strict financial sanctions imposed by the UK government.
Americans Todd Boehly and sidekick Behdad Eghbali bought Chelsea for just over £2 billion in May 2022 and sacked Tuchel 100 days later following a 1-0 defeat at Dinamo Zagreb.
Graham Potter, Steve Cooper and Julen Lopetegui are all names also thought to be being considered by the West Ham hierarchy.