A new striker was one of Palace's top priorities over the summer and, while Jordan Ayew was added on a permanent basis following his loan the season before, Hodgson was frustrated by the business done during the window.
Borini, whose contract is up in Italy at the end of the season, has over 100 Premier League appearances across stints at Liverpool, Sunderland, Swansea and Chelsea but has started just once for Milan this season and has not played since August.
The Italian, 28, joined Chelsea aged 16 and therefore would qualify as a homegrown player and would likely be brought in on loan until the end of the season, though Palace face competition from at least two Premier League rivals.
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