Sullivan, speaking at the inaugural Supporters' Advisory Board meeting at the Boleyn Ground, is understood to have told the audience that one manager has rejected the chance to succeed sacked boss Avram Grant, whilst another is set to give his answer within the next 24 hours.
Two's company: Joint chairmen David Sullivan (right) and David Gold (left) ponder their next move
Sullivan also said to have admitted that he expected the offer - a reported ?50,000 per week contract plus a ?1million-plus bonus for achieving promotion to the Barclays Premier League - to be refused once again.
Despite speculation that unsettled Nottingham Forest manager Billy Davies has been approached, bookmakers Victor Chandler make him 6-1 fourth favourite, with ex-Blackburn boss Sam Allardyce the 9-4 favourite to take over at Upton Park.
Former Newcastle manager Chris Hughton remains the second favourite at 3-1, while Martin O'Neill, who reached a settlement with Aston Villa earlier this week, and who is keen to return to management after sitting out this season, is a 9-2 shot.
Sullivan, assisted at the meeting by vice-chair Karren Brady, admitted he and co-owner David Gold offered the job to an applicant earlier this week and expected an answer from them on Friday morning.
Favourite. and the fans' favourite: Allardyce (left) and O'Neill (right) head the market for the Upton Park hotseat
No names were revealed, but Sullivan intimated that both candidates were British, and both possessed experience of managing in the Premier League or in the Championship.
Sullivan said: 'The selection of the manager is of critical importance. We are spending a lot of time on it but we have a little bit of time to make a decision.
'As you go down the list there are pros and cons with a lot of people and we are looking for someone with a proven track record in the Championship or higher or in the Premier League who wants to come to us.'
According to one insider, the Hammers are pessimistic about the chances of the individual accepting the job and Sullivan's 'gut feeling' was that they would 'turn down the job because of a particular issue' that was 'unrelated to West Ham'.
Last week a big gamble developed after speculation that Dave Jones had been approached, but Gold insisted there had been no contact with the Cardiff manager, a 9-2 contender with Victor Chandler to become the sixth West Ham manager since Harry Redknapp left the club in 2001.
Charlie McCann, PR director for the firm, said: 'Allardyce has always been a big loser with us. He was the subject of a big gamble at Christmas when there was speculation that Avram Grant was going to be sacked and we've got heavy liabilities on him.
'Martin O'Neill had been linked with the job but he's turned them down once already, but as for Dave Jones he was as short as 4-6 last week but is now out to 9-2.'
On the drift: Dave Jones (right) and Roberto Di Matteo (right) appear increasingly unlikely to be in the frame
Sullivan's comments regarding a British manager would seemingly rule out Brighton's Gus Poyet and former West Brom manager Roberto Di Matteo.
McCann also summed up wheat many West Ham fans will be feeling after Sullivan's latest comments.
He said: 'It is extraordinary that he has revealed he's offered the job to two managers. It doesn't make any sense to say something like that and would seem to be another PR own-goal.
'I'm not convinced that Allardyce would go down well with the West Ham supporters and I'd think they'd be happier with the brand of football that Poyet or Di Matteo would deliver.'
Caretaker boss Kevin Keen, who took charge of West Ham's final game of the season - a 3-0 home defeat by Sunderland - has at least one player's backing to keep the job.
Defender James Tomkins said: 'Kevin has been excellent for us. He is one person that you can say has been positive throughout the year. All the lads respect him and I have worked with him for a long time now.
'I personally think he would be brilliant for the job. Who knows who will get it, but he is a good manager and we all respect him.'
West Ham have not approached Cardiff boss Jones, insists chairman Gold Sunderland hit back at West Ham's claims of illegal approach for BaAll the latest West Ham United FC news, features and opinion
Explore more:People: Martin O'Neill, James Tomkins, Karren Brady, Chris Hughton, David Gold, Harry Redknapp, Roberto Di Matteo, Sam Allardyce, Billy Davies, David Sullivan Places: Newcastle, Cardiff