Rodgers told BBC Sport Wales he has spoken to Chelsea about the player, adding: "He's an Ashley Cole in the making."
Aanholt, 20, impressed in Chelsea's 7-0 FA Cup win over Ipswich on Sunday.
Rodgers, a former Chelsea youth and reserves coach, said: "Patrick's an outstanding player and I know [him] well. He's an Ashley Cole in the making.
"He obviously played in the last game for Chelsea against Ipswich and did very, very well so he's one of a number of names that may come into the reckoning.
"We've still got Neil Taylor out for a few weeks so yeah, it is a name that I've spoken to Chelsea on and we'll see what happens.
"We've made a number of calls already so we're just thankful that we're in a period now of games where there's not an immediate need.
"But certainly before the end of January we'd like to hopefully have another defender in to reinforce the group."
Monk suffered his knee medial ligament injury after scoring the opening goal in their 4-0 FA Cup win over Colchester United and was originally expected to be out for three months.
Monk's and Taylor's absences leave boss Brendan Rodgers short of central defensive options with Ashley Williams, Alan Tate and Albert Serran the recognised contenders.
Angel Rangel pulled out of the Colchester game with injury in the build-up leaving youngster Daniel Alfei to make his debut at right-back, with Jazz Richards filling in at left-back.
Source: BBC Sport
Source: BBC Sport