Sunderland remain winless in the Premier League relegation zone after West Ham came from 2-0 down to rescue a point on Saturday.
And Advocaat’s departure was confirmed in a statement in which he expressed his gratitude to the club.
“I want to thank everyone who has stood behind me. This is a very special football club with so many great people, but I feel it is the right time to do this – not for me, but for the club.
“I have made the decision to go after only eight games as I felt it was important to give everyone time to turn things around – like we did last year. I am thankful to the chairman (Ellis Short) for understanding my feelings and I remain on good terms with everyone at the club.
It leaves the Black Cats in a state of flux with sporting director Lee Congerton understood to be working a period of notice after tendering his resignation.
As well as Allardyce, Sunderland will consider up-and-coming managers, as they did when they appointed Paolo Di Canio and his successor Gus Poyet, with the names of Burnley’s Sean Dyche, Karl Robinson of MK Dons and Burton’s Jimmy Floyd Hasselbaink mentioned in dispatches.
Former Leicester boss Nigel Pearson is available, while there is a school of thought touting Wearside hero Kevin Phillips as an outsider.
Allardyce’s availability, proven track record and connections with the club make him an attractive proposition.
The current season is the club’s ninth since their return to the top flight under Roy Keane, and they are now looking for their seventh permanent manager or head coach since the former Man United and Ireland skipper resigned in December 2008.
They can boast only one top-10 finish during that period – they secured 10th place with 47 points under Steve Bruce at the end of the 2010-11 campaign – and have flirted with relegation in each of the last three seasons with Di Canio, Poyet and Advocaat each dragging them back from the brink.
Advocaat had tears in his eyes as Jeremain Lens struck to make it 2-0 after just 22 minutes, with Steven Fletcher having opened the scoring.
But his emotions were altered dramatically when Carl Jenkinson pulled one back in first-half injury time and Dimitri Payet levelled following Lens’ dismissal for a second bookable offence.
Pantilimon, Yedlin, Coates, O’Shea, Jones, M’Vila (Jordi Gómez 95), Cattermole, Lens, Toivonen (Larsson 84), Borini, Fletcher (Rodwell 85)
van Aanholt, Brown, Defoe, Mannone
Adrián, Jenkinson, Tomkins , Reid (Collins 66), Cresswell, Kouyaté, Noble (Zárate 79), Lanzini, Payet, Moses (Jelavic 58), Sakho.
Randolph, Zárate, Obiang, Collins, Jelavic, Antonio, Oxford