The Hammers are short of recognised full-backs - George McCartney is out of action with a long-term injury - while Daniel Potts was stretchered off during West Ham's match against Arsenal on Wednesday night with a concussion.
Joey O'Brien has filled in at left-back at times, but Sam Allardyce has tried to strengthen the problem position by moving for Bertrand - Chelsea's back-up full back.
The 23-year-old is Ashley Cole's deputy at Stamford Bridge, and although Chelsea initially considered letting him leave until the end of the season, the Blues are now thought to be keen to retain the strength-in-depth he offers.
Rafa Benitez recently revealed Cole has penned a one-year extension to his Chelsea contract, keeping him at the club past the end of the current campaign, and the European champions have been linked with Southampton defender Luke Shaw.
The Saints left-back has been tipped as a possible long-term replacement for Cole, both for Chelsea and England, which had led to speculation that Bertrand could be surplus to requirements at Stamford Bridge. But the Blues have rejected two attempts by the Hammers to bring Bertrand to Upton Park on loan.
West Ham brought in Yossi Benayoun from Chelsea earlier this season, but cancelled the arrangement last month, freeing them up to move for Bertrand. Premier League rules prevent clubs from signing two players from the same team.