Redknapp inquired about Chamakh's availability in January but the French club were reluctant to lose one of their best players mid-season.
However, Tottenham believe that a bid of around £5 million should convince Bordeaux to part with the player at the end of the season.
Chamakh, whose natural game is as a second striker in a front two, is highly thought of in France and coveted by a number of leading sides.
The 24-year-old, though, has already expressed an interest in moving to the Premier League, according to a report in the Sun newspaper.
Arsenal, too, have been monitoring the player's progress and Arsene Wenger's French connection would provide stiff competition for Redknapp if the two north London clubs were to go head-to-head over Chamakh.
Redknapp is almost certain to try to recoup some of the £16 million spent on Darren Bent while Roman Pavlyuchenko, despite recent good form, has yet to convince Redknapp of his long term future at the club.
Should Redknapp miss out on Chamakh, the manager is likely to turn his attentions to Juventus forward Vincenzo Iaquinta, who would cost in the region of £8 million.
However, Iaquinta would take some convincing of the merits of a move to England, especially to a club outside the Champions League and, in all likelihood, out of European competition altogether, despite an emphatic victory over Middlesbrough reviving hopes of a top seven finish in a troubled season.