The waters were muddied earlier on Thursday when it emerged that Graham's former club Middlesbrough had also expressed an interest, although the player's camp responded swiftly to rule out a return to the Riverside Stadium.
O'Neill's hopes of landing the striker have been complicated by Swansea's refusal to do business until they have a replacement on board, with Stoke's Kenwyne Jones among their targets.
They are yet to do that and with time running down fast, it remains to be seen whether or not they will stick to that plan.
Graham played the final 16 minutes of 0-0 draw at the Stadium of Light on Tuesday evening, and his reception was mixed to say the least.
The Gateshead-born frontman has made no secret of his boyhood affection for arch-rivals Newcastle - a move to St James' Park was never on the cards simply because he would have little sell-on value at the end of any lengthy contract - and comments made in a fanzine interview were less than complimentary toward the Black Cats.
O'Neill is desperate to boost his striking options to provide support for leading scorer Steven Fletcher, having allowed Fraizer Campbell and Louis Saha to leave the club and Ji Dong-won join German outfit Augsburg on loan.