Pars United, which the trust is part of, submitted an offer before Tuesday's deadline along with an unknown bidder, who has enforced their right to confidentiality. Fans rallied to keep the club alive after owner Gavin Masterton called in administrators in March with the club more than £8million in debt and were unaware of rival interest until recently.
Drew Main, from the Pars Supporters' Trust, said: "The apparent desire of those behind the second offer to the club's administrator to remain anonymous is a mystery and is causing not just consternation but also causing increasing concern amongst the wider fan group."
He added: "Many people are asking us why those behind the second bid are continuing to remain anonymous and we are unable to answer that question.
"It would certainly seem obvious to everyone who has been fully behind the supporters' efforts to firstly save the Pars, right through to the amazing efforts to be close to a position to buy the Pars, that these people need to show their faces and explain what their plans are for our club.
"The longer those names remain a secret the more that fans become suspicious and that is simply not good enough as everyone is working so hard to save the club we all love.
"The simple truth is that the fans have come so far and put so much into fixing the mess left behind by the previous owners that any bid without the full backing of the Pars fans is going to find it difficult to succeed.
"These concerns could lead to fans stopping buying season tickets because they are worried about who the second bid might be coming from, which could become the case very soon.
"We desperately need to avoid that happening. Right now anyone with the future of the club at heart needs to come together instead of working separately.
"We firmly believe that whoever is behind the second bid needs to come out now and tell the fans who they are and what their motivation is behind their interest, and if that means working with the fans then we need to hear about it very soon."