Bridge has been unhappy with manager Avram Grant's rotation policy this season and feels he should be first-choice left back on merit.
Despite Bridge's outstanding form this season, his frustrations are set to continue amid fears he may now be the third-choice left back behind Ashley Cole and Ivanovic.
The arrival of the Ivanovic may have paved the way for his departure, with Liverpool having already signalled their interest.
Informal approaches from Anfield have been rebuffed by Chelsea who are loathe to sell one of their first-team squad to a Premier League title rival.
Bridge, who has also interested West Ham and Tottenham and spent a period on loan at Fulham two seasons ago, is unlikely to leave in this month's transfer window but will review his position at the end of the season.