A group of 72 MPs have written to the BBC accusing it of being pessimistic and skewed in its Brexit coverage - in the months since the vote on the EU.
In a letter to director general Tony Hall, they said the BBC is "unfairly representing" Leave voters by focusing on those who regret their decision.
The MPs warned the future of the BBC "will be in doubt" if it is not seen as an impartial broker.
Lord Hall said impartiality was the "cornerstone" of BBC News.
In response, Lord Hall wrote in a letter: "We do not take it for granted and we go to great lengths to ensure that we balance our coverage and address all issues from a wide range of different perspectives."
The MPs' letter was organised by Conservative MP Julian Knight, a former BBC journalist who backed Remain but says that has not stopped him disagreeing with the stance of the coverage.
It was signed by 60 other Tories including Iain Duncan Smith and Theresa Villiers.
Two Conservative peers, three Labour MPs, eight DUP MPs, two DUP peers and UKIP's only MP, Douglas Carswell, also added their names.
They said the BBC's inability to break out of "pre-referendum pessimism" and to accept new facts was skewing its coverage of Brexit.
"If politicians and the public don't view it as an impartial broker, then the future of the BBC will be in doubt," the letter stated.
It said many Leave-voting constituencies have felt their views have been unfairly represented by the BBC.
"This phenomenon is weakening the BBC's bond with the 52% who voted Leave and all who wish to make a success of the decision made," it added.
MPs also claimed the BBC had "skewed" good economic news since the referendum and "licence fee-payers have the right to expect better".
The letter, which was also copied to the incoming chair of the BBC Board Sir David Clementi, did not include any specific examples.
It is best understood as a warning from the BBC's detractors that they expect a reasonable hearing in the Brexit negotiations in Europe, says the BBC's media editor Amol Rajan.
'Job to scrutinise'
The MPs said BBC bias can have a substantial effect on the national debate and its coverage also shapes international perception of the UK.
"We fear that, by misrepresenting our country either as xenophobic or regretful of the Leave vote, the BBC will undermine our efforts to carve out a new, global role for this country," they added.
The MPs urged the BBC to take steps to "correct these flaws" in its Brexit coverage.
In a letter addressed to Mr Knight, Lord Hall said: "Impartiality has always been the cornerstone of BBC News. It remains so today...
"It is one of the reasons why the public trusts the BBC more than any other source of news.
"I agree with you that these are consequential times. For that reason, it is more important than ever that the BBC's journalism is independent of political pressure."
Shadow culture secretary Tom Watson said: "It is the BBC's job to report the facts, not to be a cheerleader for any cause or party.
"Theresa May and Karen Bradley must make clear their commitment to the independence of the BBC and distance themselves from attacks on it by politicians who have an agenda the BBC does not and should not share."
Conservative MP and former Education Secretary Nicky Morgan, who campaigned for Remain during the referendum, tweeted that the letter was "another attempt by leading Brexiteers to silence those whose coverage and questions they don't like #chilling".