The young Englishman has struggled for minutes under manager Arne Slot and looks to be moving on for a more prominent role at Villa, who have been trying to add attacking recruits all window.
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Elliott was named Player of the Tournament for England at the U21 European Championships in the summer and won the Premier League last season with Liverpool.
The 22-year-old joins fellow deadline day signings Victor Lindelof and Jadon Sancho.