Sky Sports revealed in mid-July that fresh terms had been put to highly-rated defender Reid.
The New Zealand international only signed a new deal last summer, but that agreement will take him through to 2015.
With Premier League rivals reported to be closely monitoring his situation, West Ham were eager to ensure that Reid remains at Upton Park.
An initial offer is, however, understood to have been knocked back by the 26-year-old, forcing Sullivan, David Gold and manager Sam Allardyce back to the drawing board.
Sullivan told talkSPORT: "I think we will get to a deal with Winston.
"Both balls are in the air with regards to Winston and Ravel (Morrison) but both are players we like very, very much."