Rodgers handed in his resignation at Parkhead back in October, with Celtic icon Martin O'Neill taking over on an interim basis.
The Glaswegian giants have since been in talks with Columbus Crew manager Nancy, 48, to become their new permanent manager.
According to the Daily Mail, those talks are now at an advanced stage, and it's expected that Nancy will be in charge by the end of next week.
The Celtic job would be Nancy's first in European football, having spent all of his managerial career in the MLS.