QPR have been linked with a cut-price switch for Bramble throughout the transfer window and it is understood there is genuine interest from boss Harry Redknapp.
Bramble is in the final six months of his contract at the Stadium of Light and is expected to leave the club on a free transfer in the summer.
The 31-year-old, also a target for former club Ipswich Town, has started Sunderland’s last four games.
But a series of injury problems over the past two years will almost certainly prompt the Black Cats to release the centre-half along with fellow defender Matt Kilgallon.
O’Neill can barely afford to lose Bramble’s services despite boosting his central defensive options with the capture of on-loan Senegalese international Kader Mangane.
Wes Brown remains a long-term casualty, Carlos Cuellar is currently on the treatment table and stand-in skipper John O’Shea has suffered a series of niggling injuries in his 18 months at the Stadium of Light.
Martin O'Neill: “I don’t know anything about Titus going to QPR.
“It’s only speculation and no-one has spoken to me about it.”