In an interview with BBC television, Blair - attending his last Labour Party annual conference as leader - was asked to respond to a newspaper report that he would not publicly endorse Brown to delegates at the five-day event.
Blair, who has previously said he thought Brown would make a "brilliant" prime minister, said: "I'm not getting into the leadership contest and all of the rest of it. Gordon has been a fantastic chancellor [of the exchequer]. He's been a great servant of the country and the party. I don't resile from anything I've said but this week I'm talking to the public about the public's concerns."
"Both of us realise... that the most important thing is that this week we set out our agenda for the future. It's important the two of us work together with the rest of the cabinet for the future."
Blair, who has led Labour since 1994 and is its most successful leader, having taken the party to three straight general election victories, said it was "perfectly understandable" the media should focus on the succession.
Speculation has long been rife about the fractious nature of the two men's relations, particularly after a number of Labour lawmakers loyal to Brown rounded on Blair earlier this month, demanding he set a date for his departure.
The row, which saw eight junior government aides resign, was allegedly orchestrated by Brown, although he denied any involvement.
Blair said Sunday it was more important for Labour delegates to concentrate on the party's future direction and policy to reverse a decline in the polls to the main opposition Conservative Party.
At the same time, Brown set out his stall as Blair's successor, telling the Sunday Telegraph of the need for "new politics," with greater parliamentary and public accountability plus devolution of power to local communities.
That included changing the running of Britain's public National Health Service in the same way that he handed the power to set interest rates from the executive to Britain's central bank, the Bank of England, in 1997, he added.
In a separate BBC television interview, Brown said there would be continuity from Blair in many areas, like foreign policy and Britain's relationship with the United States, but the "Bank of England model" could be applied elsewhere.
He played down the significance of Blair's refusal to endorse him as his successor to conference delegates.
But two senior government minister - Foreign Secretary Margaret Beckett and Northern Ireland Secretary Peter Hain - both used interviews with Sky News television Sunday to publicly back Brown.
Hain, who has thrown his hat into the ring for the deputy prime minister's position, also said a leadership contest - instead of a Brown "coronation" - would be "healthy."
Home Secretary John Reid told ITV1 television that an "open, transparent" debate about the succession would be welcomed by voters, after the channel found that 81 percent did not want Brown to take over uncontested.
Polls on the eve of the conference have not made happy reading for Blair, Brown or the Labour Party. A YouGov survey said Labour had 33 percent of public support, four points down on the Conservatives.
A separate CommunicateResearch poll for the Independent Sunday predicted defeat for Labour at the next general election, which is due before 2010 at the latest.
The call for renewed party unity was reflected by ministers from Blair downwards. In his BBC interview, Blair said the focus over the next five days should be to "reconnect with the public."
By refocusing on "the public, the public's concerns and things that really worry people" -
from health, education and immigration to law and order - and adapting to the modern world, they would continue in government, he added.
"If we do that, then all the stuff of the last few weeks will be forgotten and we can concentrate on the future."
© 2006 Agence France-Presse
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