On BBC One's Andrew Marr Show, Mr Johnson said he was not sure whether presentation was Mr Brown's "strength" but added that the focus at the next election would be on "issues", such as the economy.
Discussing whether he would launch a bid for the leadership, Mr Johnson said: "I am not saying there's no circumstances."
But he added: "I have no aspiration for the leader. My aspiration was for the deputy leadership and I couldn't even get that."
He said photo opportunities and "looking good on YouTube" were "second-order issues", adding: "He can do it [the job of prime minister] and he can do it a lot better than me or anyone else."
'Speaks their language'
However, on the same programme, Labour's Mr Livingstone said: "I think Alan Johnson will probably do better at the next election than Gordon Brown... particularly in terms of recapturing our working-class support."
Mr Straw has called on Labour to "rekindle the zeal we had for power in the mid-1990s".
He said: "It is at times like these that we all need to focus on what brought us into politics, not get dragged into narrow arguments which risk the charge that we are out of touch."
And Mr Straw insisted that Mr Brown was "exactly the right person to be prime minister".
Gordon Brown has come under increasing pressure in recent weeks
When asked on whether he would run for the Labour leadership, he told Sky News' Sunday Live: "No I won't, there's not a vacancy and Gordon Brown is clearly the leader and the prime minister. He's the best man for the job."
Lindsay Hoyle, Labour MP for Chorley, said there should be less "sniping and griping" from cabinet ministers.
He added: "In fact, if you're in cabinet you have a collective responsibility and that collective responsibility is: 'Keep out of the argument, keep out of the media, let's have less comment'.
"And if you want to make comments, think about your position first. You should resign if you want to start doing articles."
Geraldine Smith, Labour MP for Morecambe and Lunesdale , agreed, saying: "What Hazel has done is inappropriate. In attacking the government she is attacking herself - what happened to collective responsibility?
"The place to raise these matters is in cabinet. If she wants to speak out and criticise, she should be on the back benches."
Former deputy prime minister John Prescott earlier urged Labour politicians who had criticised the prime minister - including former Home Secretary Charles Clarke who said recent events made him "ashamed" to be a Labour MP - to stop complaining and get behind the party.
In her article for The Observer, Ms Blears said the public did not believe many government policy announcements.
Hazel Blears said it was wrong to say she was criticising Mr Brown
She wrote: "Promote your message via YouTube if you want to. But it is no substitute for knocking on doors or setting up a stall in the town centre."
Ms Blears said Mr Brown would lead the party into the next election, but that the government must appear more "human" to defeat the Tories.
"Labour ministers have a collective responsibility for the government's lamentable failure to get our message across," she said.
In a statement later, she said: "Any suggestion that I intended what I wrote as criticism of him [Mr Brown] or his leadership is completely wrong. I fully support the collective decisions we take."
Housing Minister Margaret Beckett told BBC One's Politics Show Ms Blears was "certainly right that the [Labour] campaign is not as effective as it should be".
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