Gordon Brown will make the offer at a meeting of the UN Security Council on halting the spread of nuclear weapons and reducing existing stockpiles.
The proposed cuts come as the government searches for ways to reduce the massive deficit in public finances.
However Number 10 said keeping the UK's nuclear deterrent was "non-negotiable".
At the UN meeting, Mr Brown will call for all nations to come together to achieve the long-term ambition of a nuclear-free world.
He will say: "If we are serious about the ambition of a nuclear-free world we will need statesmanship, not brinkmanship."
Liberal Democrat leader Nick Clegg welcomed the proposals, saying they were an important development.
Clegg: 'It's an important development'
He told the BBC: "I really do welcome that finally the dam has burst on this.
"It's just unrealistic for us to believe that we can foot the £100bn like-for-like replacement costs for Trident over the next 25 years.
"I think the strategic context in which that decision is taking place is very different as well - we're not facing the Cold War threat in the same way that we once were."
The government has already announced that it has cut the UK's stockpile of Trident warheads from 200 to 160, and many Labour MPs would like the government to scrap the weapons altogether.
However Foreign Secretary David Miliband said it was imperative the UK kept hold of an independent nuclear deterrent.
He said: "We reject unilateral nuclear disarmament for ourselves precisely because the world cannot end up in a situation where responsible powers get rid of their weapons, but the danger of nuclear proliferation by other powers remains.
"As President Obama said in Prague, this is a very long-term goal which may outlive his children, not just himself."
'Useless weapons'
Officials travelling with the prime minister warned that reducing the number of submarines, which are based at Faslane on the Clyde, from four to three would not result in a proportionate 25% cut in cost, as more would have to be spent on maintaining the overall deterrent.
Kate Hudson, chairwoman of the Campaign for Nuclear Disarmament (CND), welcomed the proposals, describing them as "a serious and positive first step towards the scrapping of both the current Trident nuclear weapons system and its replacement".
But Ms Hudson emphasised the ultimate goal should be total disarmament.
US President Obama is chairing the meeting of the UN Security Council on Thursday as part of the process of drawing up a replacement for the Non-Proliferation Treaty, designed to stop countries developing nuclear weapons.
Mr Obama has said he will try to negotiate with Moscow to reduce US and Russian nuclear warheads - which make up 95% of the world's total - from 2,000 each to 1,500.
However the most pressing issue for leaders at the meeting will be how to stop the further spread of weapons to non-nuclear states.
