In an interview with the American magazine Forbes, Mr Blair was positive about the prospects for a solution, while refusing to reveal his plans.
As well as the possibility of allowing the US to use radar stations as part of their National Missile Defence programme, a report in the Daily Mail says the prime minister will allow American missiles on British soil.
But there is likely to be fierce opposition to mooted co-operation, with concerns over Britain's vulnerability and exposure to costs.
It is believed part of the missile shield would depend on the upgrading of the Fylingdales early warning base in North Yorkshire, which critics say would make Britain a prime target for anyone wanting to strike the US.
Russian opposition
Russia and China both oppose the plan, and many western European countries have considerable misgivings about it.
Mr Blair has previously refused to say whether he would agree to use of the Yorkshire base for the shield, insisting that it would be wrong to do so before any request has been received from the US.
He is due to meet Mr Bush for the first time on 23 February, when he arrives for a two-day visit to Washington and Camp David.
In an interview with the American magazine Forbes, Mr Blair was positive about the prospects for a solution, while refusing to reveal his plans.
But he said he is ready to seek a way of helping meet American desires for extra protection against missile attacks from "rogue states".
Mr Blair said any solution must also meet concerns from Europe, Russia and China about the maintenance of anti-ballistic missile treaties, which have put a brake on the arms race for more than 20 years.
Missile treaty
In the interview, Mr Blair said: "This is definitely in the box marked 'handle with care' on all sides. It is a very sensitive issue.
"I understand totally America's desire to make sure that its people are properly protected.
"I also understand the concerns people have about the ABM treaty and the desire to preserve it.
"My own judgement is that, provided we handle it with care, there is a way through which meets America's objectives and other people's concerns."
Shadow defence secretary Iain Duncan Smith, in Washington for talks with Republicans, told BBC Radio 4's Today programme the government and Nato generally had to take an active role in the programme.
"Whatever government, once they recognise the threat, must want to do something about it and work with the Americans.
"This should be a Nato programme with shared costs."
Asked about reports in the Daily Mail which suggest that Mr Blair will offer his support for the project during those meetings, Downing Street would say only: "No proposals have been put forward to us.
"The Americans are still looking at it to see how it would work."