The Black Cats have identified Moyes as the man to replace England bound Allardyce.
David Moyes and Uwe Rosler watch from the stands
The current Black Cats manager is expected be named the new England manager imminently after emerging as the leading candidate over the last week.
Sunderland manager Sam Allardyce acknowledges fans after the game
It was reported that he confirmed to his players last night that he had won the race to become the new Three Lions manager.
However, Sunderland chairman Ellis Short is demanding a hefty compensation package before he will allow Allardyce to depart.
And according to the Shields Gazette, they will only push ahead with plans to finalise a deal for Moyes when that compensation package is agreed.
The club are believed to have already held advanced negotiations with the former Everton and Manchester United manager about taking up the position.
Real Sociedad manager David Moyes watches from the stands
He is eager to return to management after leaving Real Sociedad last November, while Short has also been keen to appoint him on several previous occasions.
Sunderland will be hoping he can bring the same kind of stability that he brought during his time at Everton and help them bounce back from the disappointment of losing Allardyce.
Real Sociedad coach David Moyes is pictured before their Spanish first division soccer match against Atletico Madrid