The France U21 international hasn't featured for Leeds this season and is ready to quit Elland Road.
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Tuttosport says Meslier's agent Pini Zahavi is trying to secure a transfer to Serie A.
Inter Milan and AC Milan are interested and Zahavi is working on placing Meslier at one of the two Italian giants.
Both Milan clubs have a good relationship with Zahavi. The biggest problem is that Leeds want to sell the goalkeeper in January, while Milan and Inter would prefer to secure a Bosman transfer in the summer.
Milan look set to lose Mike Maignan, 30, as a Bosman this coming summer, while Inter are looking for a replacement for Yann Sommer, 36.