They were due to come into service in 2014 and 2016, as the biggest and most powerful warships the UK has ever built.
Much of the work is due to take place at Scottish shipyards on the Clyde, where 4,000 jobs are dependent on the project.
Costs to soar
The project's cost is now expected to rise from £3.9bn to around £5bn.
Armed Forces Minister Quentin Davis MP: 'We have to live within our annual budget'
The memo was written for the chief executives of the companies participating in the project and attributed the cost increase to "a combination of direct costs, inflation and accounting adjustments".
"The MOD [Ministry of Defence] will publish its annual report and accounts in July; these will show £1bn of QE Class cost growth and the project will come under severe pressure through the opposition and the media," the memo said.
"This is a very real fight for the programme's survival," it adds.
The memo also suggests the future of the Appledore shipyard in Devon could be under threat and possible measures to reduce costs include "substantial redundancies" of the order of 400 to 500.
HMS QUEEN ELIZABETH AND HMS PRINCE OF WALES
Displacement: 65,000 tonnes
Length: 280m (920ft)
Width (at flight-deck level): 70m (230ft)
Keel to masthead: 56m (184ft)
Nine decks (plus flight deck)
Speed: 25+ knots
Range: 8,000-10,000 miles
Aircraft: 36 F-35 Joint Strike Fighters and four Airborne Early Warning aircraft, plus EH 101 Merlin helicopters
Crew: 1,450 (including air crew)
Weapons: Phalanx close-in weapon systems; 30mm and mini-guns
Source: Ministry of Defence
The BBC's business editor Robert Peston said the increase in costs would be seen as alarming, especially at a time when there are intense pressures on the government to cut public spending.
He added that if the project was scrapped, the knock-on effects would be serious.
For example, some 80,000 tonnes of steel worth £65m has been ordered from Corus, the beleaguered Anglo-Dutch steelmaker.
A consortium of companies are involved in the project including BAE System, VT Group, Babcock and Thales.
BAE Systems said that the delay to the project announced earlier was always expected to lead to an increase in costs.
The work is split between various yards, including Portsmouth, Rosyth, and Hebburn.
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