Speaking to the Independent, the unnamed individual's comments reflect an apparent growing unease amongst the Labour ranks that their leader won't be able to deliver a win in next year's general election.
Miliband has said he remains determined to attract voters "from every walk of life" as he seeks to quell criticism of his party's local election results.
The Labour leader is to confront the party's most prominent failure by returning to the campaign trail next week in a key seat where it lost control of the council.
A surge of support for Ukip in the Essex town of Thurrock - which is second on Labour's 2015 target list - saw it take two Labour and three Tory seats and leave no party in overall control of the authority.
Thurrock had been hailed by Mr Miliband when the party took power there last year as evidence that the Opposition was "winning back trust, gaining ground".
Despite winning more than 300 extra seats in the elections, the party failed to make the sort of advance in some key areas seen as vital to securing an overall majority in the Commons next year.
A projection of national vote share on the basis of the local votes suggested Labour was on 31% - just two ahead of the Conservatives and not on course for a dramatic return to power after being ousted in 2010.
But the Labour leader pointed to other areas where the party had performed strongly as evidence he could secure the keys to 10 Downing Street.
"The local elections show Labour can win because it is our party which is winning where it matters in dozens of our target seats for the next election," he said.
"From Cambridge to Redbridge, from Crawley to Amber Valley, people are electing Labour councils to meet their desire for change."
A new poll of marginal seats also gave Labour some encouragement, giving it a 12-point lead over the Tories - enough to oust 83 Tory MPs and produce a healthy majority at Westminster.
"For too long, millions of people have felt locked out of our economy or let down and ignored by politics.
"There is a deep sense of discontent with the way our country is run. I am determined to show people, including those who voted for Ukip, that we can change our country so they can build a better life for themselves and their family.
"In the weeks and months to come you will see our Policy Review finalising our programme for changing our country so we can connect the wealth of our nation to the lives of working people once again.
"Some will tell you we only need the votes of certain people to win. I say we can't build a better country unless people from every walk of life know there is a future for them if they work for it.
"This is at the heart of what I mean by One Nation. This is the mission for One Nation Labour.
"This is why we will campaign as we mean to govern - as a One Nation party reaching out to everybody."
An opinion poll underlined the personal unpopularity with voters Mr Miliband must overcome - showing more people backed Labour despite his leadership than because of it, by 36% to 31%.
He is the only one of the main party leaders to suffer a negative rating on that score, with Conservative Prime Minister David Cameron an appeal to 48% of his party's voters and a negative to only 18%.
The Survation survey for the Mail on Sunday found Labour would be more attractive to voters with a new leader by a margin of 26 points, twice that for the Tories.
A lack of charisma and "common touch" were the main objections to Mr Miliband made by the 1,107 adults asked to give their opinions
They ranked shadow business secretary Chuka Umunna as their favourite to be the next Labour leader - with London mayor Boris Johnson the pick of the Tories and Energy Secretary Ed Davey for the Lib Dems - after being shown videos of politicians and asked to rate them on a number of attributes.
Labour former cabinet minister Frank Field said the election result "poses big questions for the Labour leadership as a whole" - complaining the party had been "silent" on immigration.
"Ukip poses a huge threat to us. We have got quite a lot to do," he told the Sunday Telegraph.
"The danger for Labour is that our supporters have been more affected by immigration than any other group in the country and we have been more silent.
"We ought to be thinking about what we learn from Ukip's campaign. How we put issues that concern people in a language they speak themselves ought to be our major concern."
A lot of the party had been "in denial" on the issue which pushed voters to Ukip, he said.
"They have chosen to say what they didn't like, which they couldn't do through the traditional parties. We have got the most mega lessons to learn."
Labour deputy leader Harriet Harman said: "Three hundred new councillors would be baffled by the idea that we should think that we all had to be turning inward and criticising ourselves.
"As far as the public is concerned, us turning inwards and having self-criticism is not a good idea. We need to be listening to their concerns and acting on them," she told BBC1's Andrew Marr Show.
She played down the significance of "every little nuance" of polls showing Ed Miliband is a less popular choice as prime minister than David Cameron and dismissed projections based on the local results which suggested Labour was off course for a 2015 victory as "for the birds".
"The facts are we are moving forward. The facts are Ed Miliband is in touch with people's concerns and that we are putting forward those policies.
"Since 2010 when Ed Miliband became leader we have had the momentum of moving forward when people are voting. The momentum is there."