The consortium, led by Azerbaijani businessman Nasib Piriyev with former Queens Park Rangers chief Philip Beard on board, say they have seen two attempts over nine months pushed back by Sullivan and have now accepted a deal cannot be done.
West Ham maintain that they have never received a formal offer from the group.
PAI have stressed that if circumstances change they would remain willing to discuss another takeover attempt, but Sullivan has always said he would not sell to the group and according to a statement from PAI feels this is not the time to sell with the Hammers on a high under David Moyes.
The firm say the talks over a second approach were "amicable" after earlier attempts result in an ugly and public war of words between the two parties, with Sullivan branding the offer "derisory" and PAI Capital as "property developers" given their interest in the Olympic Park.
Taking ownership of the club's stadium was one part of PAI's plan and they had held discussions with the London Legacy Development Corporation, the group who manage the stadium, over buying the London Stadium. PAI also claim that the second offer would have seen an injection of £150million into the club for investment in "the training ground, youth academy, scouting set-up and player purchase across the men’s and women’s teams."
PAI had brought former West Ham players Tony Cottee, Rio Ferdinand and Anton Ferdinand on board to back their bid.
In a statement, PAI said: "Earlier this month, PAI Capital submitted a further offer for West Ham United to the controlling shareholder, David Sullivan. The offer was well-received and discussed on an amicable basis under agreed confidentiality.
"Mr Sullivan has, unfortunately, declined the offer on the basis that, with the team performing so well, it is not the right time to sell. PAI Capital notes his decision and, given that it is one of timing and not price, the company is unable to take things forward at this stage. PAI Capital’s interest remains however and, should things change, will be ready and willing to proceed."
Piriyev, PAI’s managing director, said: “I am disappointed that we cannot conclude the deal as hoped, as we have some big ideas for the club. This is something we have been working on for over nine months, with a committed team behind the bid.
“West Ham is close to my heart and I hope that our interest has been healthy for the club - I believe that all competition is good for focusing minds and improving performance.
“Of course, we respect the owners’ wishes and simply put on the record that we are here if, and when, they decide the time is right for them to sell. Until then, I wish all the best to the club, and I thank the fans for their patience, understanding and support.”