Boro are keen to offload O'Neil during January and have a valuation that has not been met by any of the many interested clubs so far.
The Hammers and Wolves are the latest to have made their interest known, but Middlesbrough hope to receive around £2m for a midfielder with 18 months remaining on his contract.
Bolton, Everton and Stoke have also been credited with interest in acquiring the midfielder who has admitted recently that he is no rush to turn his back on Teesside. Aston Villa and Birmingham City are also pondering whether to make a formal offer, but his departure is far from certain.
West Ham have been in contact with Middlesbrough chief executive Keith Lamb but were unable to reach an agreement for a player that cost £5m when he moved from Portsmouth in August 2007.
That has forced the Hammers to step up their pursuit of alternatives, with Aston Villa midfielder Steve Sidwell expected for advanced discussions at Upton Park before Christmas.
ONeil's Premier League wages and transfer value make him the club's most aleable asset, with the other, David Wheater, out of contract and available for nothing at the end of the season.
Moving those two players on at this stage represents Mowbrays best chance of raising his transfer funds to embark on his own squad restructuring.
He has targets in mind, but sees little point in pushing things too far when he will only have cash available to spend by reducing the wage bill and bringing in some money from the sales of players such as O'Neil and Wheater.
Mowbray is also looking for Kris Boyd to leave, just six months in to his career at the Riverside Stadium, but accepts that Middlesbrough will do well just to move him on for free as one of the squad's high earners.
O'Neil is a player Mowbray would prefer to keep. With relegation looking more likely than promotion this season, however, a realistic offer is likely to lead to his departure.
Middlesbrough know lining up deals now is crucial, given the longer it takes for the money to come in means it will take longer for Mowbray to start to put his own stamp on the squad.