The unsettled Swansea frontman wants to return to his native north east - he is from Gateshead - and has been strongly linked with a switch to the Stadium of Light.
He came on in the goalless draw on Tuesday and almost snatched a late winner for the visitors and O'Neill revealed afterwards that the Black Cats had made "a couple" of offers and were hoping for a breakthrough before Thursday's deadline.
He said: "We have made bids for a couple of players, but we didn't make them overnight, we made them some time ago.
"Of course it's absolutely the prerogative of the team to whom we have put these bids in either to accept or not accept as the case may be.
"We wouldn't be the only club in that sort of position, this idea that we are waiting until January 31.
"There may be a host of signings in the next 48 hours, I don't know, but there's every possibility because that seems to have been the way of it in the last number of years."
Sunderland last week