A year since his arrival from Juventus, Chiesa could be on the move again this summer.
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A return to Italy has long been mooted for Chiesa, where AC Milan and champions Napoli are both interested.
However, the Mirror says Fulham and Celtic want to keep Chiesa in the UK this season. Both clubs want to take the Italy international on-loan for the new campaign.
Chiesa, 27, missed Liverpool's Asian preseason tour with management insisting it was due to a slight injury. However, his absence also fueled talk last month of a move to Atalanta, though any move hinges on the potential sale of Ademola Lookman.