The Hammers have been rebuffed in their first offer but they have asked Sunderland to name their price for the 34-year-old English striker, Sky Sports has reported.
Signing Defoe however might not be such an easy task for West Ham, as Sunderland have so far been in dire need of his scoring. The veteran striker has been on target 11 times so far this season, having scored more than half of Sunderland's 19 goals.
West Ham on the other hand have relied on the production of Michail Antonio as both Andre Ayew and Andy Carroll have struggled with injuries and have combined for a mere three goals.
Meanwhile Italian international Simone Zaza is rumoured to be on his way out of West Ham in the January window, making the acquisition of a striker a matter of imperative importance for Hammers manager Slaven Bilic.