The English defender – who is also wanted by Everton – is some way down the pecking order at Stamford Bridge, with Jose Mourinho having elected to field Cesar Azpilicueta – a right-sided defender – in the role after dropping Ashley Cole.
This has alerted Rodgers, who is keen to bolster his squad of the coming month.
A left-back is seen as a priority for the Northern Irishman, who was forced to field centre-back Daniel Agger in the position due to injuries to both Jose Enrique and Jon Flanagan.
Bertrand, 24, is believed to have little future at Chelsea, but Mourinho is unsure as to whether or not to sell to a direct rival, particularly with the West London side having witnessed Daniel Sturridge’s rise since swapping Stamford Bridge for Anfield in 2013.
Liverpool may indeed be expected to pay a higher fee than some other interested parties, but Rodgers could be willing to take the hit for the-two time England international.
Everton are another side interested, but their willingness to conclude a swoop is dependent on the future of Leighton Baines.