The left-back barely featured for the Blues at the start of last season before he sealed a loan switch to Aston Villa in January in search of first-team action.
With Ashley Cole set to be released, there has been talk of Bertrand heading back to Stamford Bridge to act as cover for converted right-back Cesar Azpilicueta, yet the Londoners are thought to see the 24-year-old as surplus to requirements.
Liverpool were understood to be leading the race for the defender, but now Spurs are firmly in the battle.
Mauricio Pochettino was believed to have been targeting a reunion with Luke Shaw at White Hart Lane, but with the England international valued at £30m by Southampton, a deal is unlikely.
The Argentine sees £6m-rated Bertrand as a better option, and is keen to convince him to make the short trip from Chelsea to Tottenham.
The left-back role was a problem for the Lilywhites last term, with their only genuine option, Danny Rose, having struggled with injury woes and inconsistent form.
Kyle Naughton and Jan Vertonghen often filled in, but neither player looked at home in the position.