Hammam stated the radical change had to be made to make ammends for "our financial mistakes of the past."
Jones has been told to cut
While the club need to raise £2m as quickly as possible, most likely through the sale of captain Danny Gabbidon to a Premiership club.
Players having left Ninian Park in the last week are Lee Bullock who joined Hartlepool, while top scorer last season Peter Thorne joined Norwich City. Bids have been made for Chris Barker from Nottingham Forest, Danny Gabbidon and James Collins (both West Ham) while Millwall are keen on goalkeeper Tony Warner.
Jones knows he has to ship players out to bring in the new men he wants to turn around the club's fortunes on the park.
Hammam says the plan will be for Jones to have a 23 or 24-man squad to lead the Bluebirds into the new campaign.
Cardiff's money problems stem from delays in their new stadium project, which could guarantee them revenue, however, until that begins to move forward thhe club must slash all other major expenditure as well as trying to pay off various debts.
The Bluebirds are believed to owe the Inland Revenue and a six-figure sum needs to be paid to Rotherham for their signing of Alan Lee..
Hammam said:"Dave's come into the job here with me, the fans and you demanding that we get in the play-offs. And suddenly he has to achieve that with these cuts I am asking of him.
"Dave's going to need the intelligence of Einstein, the power of Hercules and the luck of the Irish to do what we are demanding."
He added though, "That said, I have to say Dave has been very calm about the whole thing. He had a fair idea, when we originally spoke to him, of what we needed to do.
"We've discussed the detail of it and his attitude is one of, 'Oh, that's what is needed - leave it to me. I will try to get in player X, Y or Z who I think can help us, and maybe we can make cuts with player A, B or C.'
"His attitude is so commendable, considering the circumstances. I'm telling him to cut the budget by a third, I'm telling him to also generate £2m.
"And yet, within that budget, Dave knows he still has to have players who are on big wages who might not form part of his plans.
"I'm sorry for him, I'm asking something very difficult of him. But I would sum it up by saying I'm giving him very little choice, rather than no choice whatsoever.
"If anyone can cope, Dave can. Because I sense I'm talking to an organised, calm man whose attitude is one of, 'Fine, you want me to do this. Well this is how I intend to do it.'"
Willie Boland has signed a new deal with the club, although manager Jones has decided against re-signing Darren Williams.
"To get players in, we have to get players out," Hammam continued.
"Look, let's be straight about this. We have to change for changes sake, because this team has stood still - and that's a side that has included Earnie and Graham Kavanagh.
"Clearly, it wasn't a team that was going to get us promotion, so Dave will be looking to bring in the men he wants.
"But the bottom line is that we have to cut the wage bill by a third from what it has been.
"That said, it has to be pointed out that the budget Dave will be working to is still pretty good, in comparison with most sides in our division.
"Teams that have come down from the Premiership, like
"But we can still compete with most clubs at our level. It's not a bad budget, by any stretch of the imagination.
"The problem we've got is that we're working against the clock, as Dave looks to change the squad around in time for the new season.
"But managers have just started reporting back for pre-season training, so they have begun picking up the telephone and speaking to one another as we talk."
Speaking about the delay to the new stadium, Hammam said: "Until the developer gives the final go-ahead, we can't sort out the short-term finances.
"Once the stadium is signed, sealed and delivered, we can get on with planning the football properly. But, as that has still not happened yet, we have no option but to make these cuts.
"What we're doing is paying for the mistakes we've made in the past. And, as the leader of this club, I take responsibility.
"I won't shirk anything. I am leading things for the club, the fans, Welsh football and the media. Surely, for example, if we are playing
"So we're in this together and I fully realise my responsibilities. I'm angry with myself that I've let it get to this, but we've made decisions ... and we move forward from here."