According to Sky Sport Italia, the Premier League club sent a written proposal late Friday evening.
They are eager to take the former Juventus hitman on loan, paying an undisclosed sum for the privilege.
Milan have been trying to off-load Matri in the January transfer window after he scored just one goal in 18 appearances between Serie A and the Champions League.
The 29-year-old only arrived on August 30 for €11m (£9.6m), but failed to make an impact.
West Brom have also been linked with Roma striker Marco Borriello in a similar loan deal.
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