Sources close to Prime Minister David Cameron said that it was down to Entwistle's "conscience" whether he accepted the £450,000 pay-off - representing a year's salary - despite him serving only 54 days in the job.
But the BBC Trust has insisted the settlement "was justified and necessary".
Entwistle quit on Saturday (November 10) after a Newsnight report earlier in the month led to former Tory treasurer Lord McAlpine being wrongly accused of child abuse.
According to Number 10, Cameron feels that it is "hard to justify" the pay-off to the former director general, but the prime minister is taking a "hands off" approach to the matter.
But Labour has branded the pay-out "not justifiable", and has secured permission to ask an urgent question in the Commons at 3.30pm today.
Culture secretary Maria Miller said that the BBC Trust must give justification as to why Entwistle should get double the six-month severance payment that he is entitled to under his contract.
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"This is a large amount of money, and tough to justify considering the circumstances of Mr Entwistle's departure and his contractual arrangements," she said.
"The Trust will need to justify this - it is accountable to licence fee payers in ensuring value for money, and we expect it to have considered that carefully."
However, the BBC Trust has said that the size of the pay-off to Entwistle reflected the fact that he would continue to support the BBC, including on two major inquiries into the Jimmy Savile sex abuse allegations.
BBC Trust chairman Lord Patten has also written a letter to John Whittingdale MP, the chair of the Commons culture committee, in response to criticism of the Entwistle pay-off.
He says in the letter that "a settlement on these terms was justified and necessary".
"In circumstances where we needed to conclude matters quickly and required George's ongoing co-operation in a number of very difficult and sensitive matters, including the Inquiries into issues associated with Savile, I concluded that a consensual resignation on these terms was clearly the better route."
He adds: "I consulted my colleagues on the Trust's Remuneration committee and took legal advice. Our conclusion was that a settlement on these terms was justified and necessary. The alternative was long drawn-out discussions and continuing uncertainty at a time when the BBC needs all of its focus to be on resolving fundamental issues of trust in BBC journalism."
Tim Davie, the acting BBC director general, has said that the Entwistle pay-off is a matter for the Trust to address.
Davie stressed that "an honourable man" had left the BBC, referring to Entwistle, but noted that his job is now to get a "grip" on what is happening.
Speaking to BBC News, Davie also said that the most recent events must not overshadow the serious allegations of child abuse involving Savile that are currently being investigated
"We shouldn't let events take over what are serious allegations against Sir Jimmy Savile, which is the bigger issue," said Davie, who was named the new boss of BBC Worldwide before this controversy broke.
Also today, BBC director of news Helen Boaden and her deputy Stephen Mitchell have "stepped aside" pending a review of Newsnight's decision to drop a report into the Savile allegations last year.