Stefano Pioli's side were dealt a heavy blow last weekend afterKjaer, 32, injured his cruciate ligament, ruling him out for the remainder of the season.
Milan, led by sporting director Paolo Maldini, are on the hunt for cover this January and Sky Sport Italia suggest they have identified Kjaer's Denmark team-mate Christensen as a potential option next month.
Christensen, 25, has fallen down the pecking order at Stamford Bridge this season and is in the midst of a deadlock with club chiefs over a contract extension.
The Dane's deal expires next summer and Chelsea were reportedly left stunned when the 25-year-old asked for the club's five-year offer to be reduced to three years.
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