Elliott is ready to leave the Premier League champions this summer in search for regular first team football.
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Major interest from RB Leipzig has arrived for Elliott, though a return to London is also emerging as an option.
The Sun says the former Fulham junior is interesting West Ham.
The Hammers plan to ask for a loan-to-buy arrangement with Liverpool for Elliott, as they cannot afford to meet the Reds' £40m valuation.
For his part, Elliott is keeping an open mind, with a move abroad not being ruled out.