Leeds and West Ham agreed an undisclosed transfer fee for the 26-year-old on Monday and Bowyer then travelled to London to discuss a contract with Upton Park officials.
"We had a very good meeting and when the paperwork is completed, as we are sure it will be, we are confident the deal will be done in the next 24-48 hours," West Ham managing director Paul Aldridge told his club's official website late on Monday.
West Ham manager Glenn Roeder added: "Everything went well and we expect things to be finalised sooner rather than later."
Providing Bowyer passes a medical, bottom-of-the-table West Ham hope to have the 26-year-old in their squad for Saturday's English premier league clash with Newcastle United at Upton Park.
Bowyer's Leeds contract expires at the end of the season and he has rejected a new deal at Elland Road, where he has been unsettled since his involvement in a high-profile court case in 2001.
Last July a proposed nine million pounds ($14.44 million) move to Liverpool fell through but British newspapers on Tuesday said West Ham were likely to pay Leeds as little as 250,000 pounds for Bowyer.
Leeds, who are looking to trim their wage bill, are keen to sell Bowyer during January's transfer window because he will available for nothing when his contract expires at the end of the season.
