The Teessiders confirmed that they have given the Barclays Premier League club permission to talk to the 27-year-old.
A statement said: "Boro have accepted an undisclosed bid from Premier League club West Ham for midfielder Gary O'Neil. The Hammers have been given permission to discuss personal terms with the former Portsmouth man."
O'Neil joined Boro in a £5million switch from Portsmouth in August 2007 under former boss Gareth Southgate.
He initially struggled to settle in the north-east and was linked with a return to the south coast on several occasions. But he eventually found his feet and was appointed captain as he established himself.
However, Boro's relegation from the Barclays Premier League and failure to win promotion at the first attempt have put ever greater pressure on the club's finances, and with little sign of a push for the top flight materialising this season, purse strings are having to be tightened.
Source: PA
Source: PA