It is understood that the members of the boardroom oppose the hiring the former Newcastle manager because of his association with their rivals and his rejection of their advances before they hired Roy Keane in 2006.
The aversion to Allardyce in the corridors of power at the Stadium of Light means they could step up their interest in Celtic manager Gordon Strachan's services although any overtures are expected to be strongly resisted by the Glasgow club.
On Saturday Athole Still, Strachan's agent, told Telegraph Sport there had been no contact between the two parties, stating: "There has neither been an approach from our side nor any approach from Sunderland. It's just speculation. Last week it was Sven-Goran Eriksson." Former West Ham manager Alan Curbishley also has his admirers in the Stadium of Light boardroom even though he would fail to bring the "box office" appeal that the club craves.
Two potential high-profile candidates who would fit the bill in that respect are: Bernd Schuster who surprisingly signalled his interest in the post following his sacking as Real Madrid manager and Brazilian legend Zico, who is currently managing Bunyodkor in Uzbekistan.
Despite opposition to Allardyce, on Sunday he was still the bookies' favourite to succeed Keane but remains on tenterhooks.
"There's been no contact between me and Sunderland," Allardyce said. "But I'd like to talk to them and we could see where it went from there. I'd like to get back into the game. I'm raring to go."
Sunderland caretaker manager Ricky Sbragia has insisted he will not apply for the position despite this handsome victory.
Sbragia said: "I've never ever thought about being a manager. I've always liked being a coach. I'm in a wonderful job but I've never spoken to Niall Quinn [Sunderland chairman] about the manager's job. It's just about trying to get a win."
West Brom captain Jonathan Greening, who worked under Sbragia during his coaching days at York, won't be putting any money on him to succeed Keane.
"Some caretakers say they don't really fancy the manager's job and don't mean it but he does," Greening said. "He genuinely prefers being a coach."