David Cameron has not spoken to Lord Patten since the resignation of George Entwistle, the Prime Minister's official spokesman said.
Culture Secretary Maria Miller "is leading the contacts" between the government and the BBC Trust.
The Prime Minister's spokesman said George Entwistle's £450,000 pay-off was "hard to justify", but that it was a matter for the BBC Trust to decide.
David Cameron has full confidence in Lord Patten as chairman of the BBC Trust, said the Prime Minister's official spokesman.
He added: "The important thing is for Chris Patten to lead the BBC out of its present difficulties."
The Culture Secretary Maria Miller will speak to Lord Patten about the issue of George Entwistle's pay-off.
This is a large amount of money, and tough to justify considering the circumstances of Mr Entwistle's departure and his contractual arrangements. 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.
The BBC is a global British institution, and above all else it is vital that it focuses on restoring its credibility. The Trust needs to act swiftly to ensure that the management and leadership issues in the Corporation are resolved.
– Maria Miller, Secretary of State for Culture, Media and Sport
I felt that the Trust was slow off the mark in responding to the crisis but that it is now acting decisively. We must not lose sight of the Inquiries at the heart of all this. They remain of the utmost importance.
Harriet Harman MP, Labour's Shadow Secretary of State for Culture, Media and Sport, said:
It is not justifiable for the BBC to pay double the contractually required sum to the director general on his resignation. It looks like a reward for failure.
George Entwistle should decline to accept any more than is required under his contract.
This is not the way to restore public confidence in the BBC.
Pressure is building on Lord Patten in reaction to the news that the former BBC Director-General will receive a full 12-months salary for 54 days in the post - despite his contract only entitling him to 6-months.
There are now questions being raised over Lord Patten's future as Chairman of the BBC Trust:
The Daily Telegraph are reporting that in addition to a full years salary for his 54 day tenure as BBC Director-General, George Entwistle will also receive his full pension, bringing his total payout for the job to £1.32 million.
His £450,000 yearly salary would mean that Mr Entwistle was paid £8,333 a day as the top BBC boss and before receiving a pension pot, reported by the Telegraph to be £877,000.
MPs from both sides of the House have taken to Twitter over the news that George Entwistle will receive a full year's salary after he resigned as Director-General of the BBC:
Big mistake for Entwistle to be awarded such a large package. This highlights that #BBC Trust & management is out of touch
Entwistle gets one year payoff after just 53 days in the job. Unbelievable.
£450,000 for 54 days work. If George Entwistle resigned why is he entitled to a full year's salary? May be standard at BBC but anywhere else
The chairman of the Commons Culture, Media and Sport Committee John Whittingdale said he could not see any justification for George Entwistle to receive such a large pay-off.