The England international has been considering his options at Liverpool this summer.
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Elliott has doubts about where he sits in manager Arne Slot's plans and has been exploring his options across Europe.
The Sun says Palace have now made a move and opened talks with Liverpool for the midfielder.
The Reds are prepared to sell - at a starting price of £40m. Any sale will boost Liverpool's funds to go towards their bid for Newcastle's wantaway striker Alexander Isak.
A swap involving Palace skipper Marc Guehi also cannot be ruled out, with Liverpool in talks for the defender.