Potential buyers of the financially stricken club have until 12 July to submit offers to administrators BDO.
Murray remains pragmatic on the situation, but believes his group can still push a little further in their efforts to take control at Tynecastle.
"Being the preferred bidder is not in our hands," he said. "That's for BDO and all we can do is put together the best possible, and most credible, bid for the club.
"We can only do that with the support of the fantastic and phenomenal fans at this club who've been converting their pledges to their direct debits.
"Now we still need more because the more direct debits we have converted, the more we can persuade the administrators that we should get preferred bidder status to be able to take forward the purchase of the football club.
"But we're very, very confident that our bid will go in. It will be a very good bid. It will be a very credible bid.
"And then it's up to the administrators to decide whether or not they want the Foundation of Hearts to purchase this football club."