Boro are keen to offload O'Neil during the January transfer window 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 two clubs 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 or not 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.
O'Neil's Premier League wages and transfer value make him the club's most saleable 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 manager Tony Mowbray's best chance of raising his transfer funds to embark on his own squad restructuring in January.
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 like 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 in an ideal world. With relegation looking more likely than promotion out of the Championship 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.
At the start of this year they were caught out on deadline day when they reluctantly accepted an offer from Manchester City for Adam Johnson, sparking last minute swoops for Scott McDonald and Lee Miller.
One player that Mowbray will not be selling next month, though, is Jason Steele. The young goalkeeper has attracted attention from Premier League clubs including Arsenal this season, but last week's decision to sign a long term contract brought an end to talk of a January transfer.
And while Mowbray is keen to keep his best players, he would prefer more top-flight clubs to be tracking the members of his squad because that would at least highlight he has a squad full of talent at his disposal.
As things stand, with Middlesbrough spared a place in the bottom three by goal difference and a lack of form running through the team he has inherited, he would struggle to find a buyer for most of his players.
"From our point of view, this has been the case at Hibernian, West Brom, I think it is good that you get footballers other people want," said Mowbray's long term No 2 Mark Venus.
"I would like to hear that there are bigger teams chasing all of our 20 players because that means we would have a decent team, but that is where we are.
"How footballers handle it is a different matter, but I don't want to go away. If people come and bid a lot of money and take them away then the law of the jungle says we get some money and they may become better players. That's life. You have to be fair and open minded."