Rodgers is assessing his squad at the Madejski Stadium after recently taking over at the Championship club.
The manager is a huge admirer of Cork having worked him with during his time at Chelsea and Watford, and believes he would be a valuable addition to his side, skysports.com understands.
Cork enjoyed a spell on loan at Watford last season and impressed during his time at Vicarage Road.
The 19-year-old is due to report back to Chelsea for pre-season training next month and he is waiting to see if Chelsea plan to loan him out again for next season.
Rodgers is hoping to use his connections at Chelsea to lure Cork to Reading, seeing the England Under 21 international's versatility as an attractive proposition for his new-look Royals next season.