The Leeds chairman, Peter Ridsdale, last night vowed to emulate Spurs' Alan Sugar in capturing O'Neill and, as he revealed in Saturday's match programme (against Leicester), that involved four approaches after an initial rebuff.
"I am certainly not going to give up after one telephone call," Ridsdale said. "I spoke to Alan Sugar a number of times before the Graham situation was resolved. It may be that I end up doing something similar."
"I'm not downcast. I'm very pleased that we made a speedy resolution at board level and I'm pleased that it was unanimous. Now we target our man and I hope I'm as successful as Alan Sugar on two counts - firstly, he did it within the rules and secondly, he got his man, and that's what we intend to do."
Leeds did not say O'Neill, 46, was their choice but Filbert Street later revealed the news in a terse statement to the Stock Exchange. "Leicester City Football Club confirm they are the club approached by Leeds United in respect of their football manager, Martin O'Neill," it read. "The boards of Leicester City PLC and Leicester City Football Club are resolute in their refusal to allow Leeds United to speak to [him]."
However, O'Neill, who was last night in talks with the chairmen of Leicester City football club and its plc, John Elsom and Sir Rodney Walker, only has to say he wishes to see what Leeds have to offer - just as Graham did with Spurs - to set the wheels in motion.
His contract, which has two years to run, is understood to contain a clause guaranteeing his release on the payment of pounds 1m in compensation. That figure falls to around pounds 300,000 after 31 October, which may persuade Leicester to release him.
There have also been suggestions that O'Neill is less than happy at Filbert Street. Last summer he asked for financial commitments from the board in terms of buying players before turning down an offer from Everton, and on Saturday his comment that he felt he was "reasonably popular with the fans" did not convey an impression of perfect harmony.
"You can turn things down for so long but it gets to the point where people stop asking you," Casey Keller, Leicester's goalkeeper, said yesterday. "I don't want to speak for [O'Neill] but if Leeds ask him and the deal is right I assume he will go, and so do most of the players.
Leeds are hardly likely to be deterred by financial demands from either Leicester or O'Neill. Graham was reportedly the highest-paid manager in the Premiership at Elland Road, while they received around pounds 3m in compensation from Tottenham. The Leeds board, which has spent pounds 23m in just over two seasons, has also committed itself to providing the new manager with new funds to buy players.
Certainly Ridsdale's relaxed attitude yesterday suggested he was confident he would get O'Neill, probably before Leeds' next game, against Nottingham Forest on 17 October. "I will sit and reflect on the situation," he said. "That's exactly how Tottenham did it and I'm hoping the other club I am talking to will also sit and reflect. They know where I am if they change their position. They know where to reach me.
"I would very much like to get our man, but given that the person concerned is currently under contract, it's out of my hands. We're talking about a long-term appointment, we don't have to rush and we don't have to worry too much. We will see."
Ultimately, contract or no contract, it will depend on how much O'Neill wants to join Leeds. For the time being, David O'Leary and Eddie Gray remain in charge of first-team affairs although whether they will survive the new appointment is debatable.
O'Neill is almost certain to bring in the backroom staff who work with him at Leicester - John Robertson, Jim Melrose and Steve Walford - which would place both O'Leary's and Gray's positions under threat.
There is a vacancy as Graham's assistant at Tottenham, of course, and the post of manager at Filbert Street might also interest O'Leary should it become available.
The managerial merry-go-round keeps on turning.