Paul Scully told the BBC Boris Johnson had apologised and would now “rebuild trust with people who are angry”.
The PM is working to secure support from his own MPs, ahead of a vote on whether a committee should investigate him for misleading Parliament.
The vote, secured by Labour, will take place on Thursday.
Labour has accused Mr Johnson of lying to Parliament about gatherings in Downing Street during lockdowns – something he has denied.
Under government rules, ministers who knowingly mislead Parliament, which the PM denies doing, are expected to resign.
If MPs vote for an inquiry, the privileges committee – made up of seven MPs – could recommend sanctions, including an apology, a suspension or even expulsion from the Commons.
But the move is unlikely to succeed because the majority of Conservative MPs are standing by the PM, and are likely to be ordered to vote against the Labour motion.
Mr Scully said the government would wait to see the wording of the motion before deciding how to vote.
Speaking on BBC Breakfast, he said Mr Johnson “wants to move on with the really important issues of the day”.
“That’s difficult because he has to rebuild trust with people who are angry and frustrated but that’s the challenge that we have.”
On the same programme, Labour’s deputy leader Angela Rayner reiterated her party’s call for Mr Johnson to resign.
She said Mr Johnson was “undermining our democratic processes” and “debasing the office” of prime minister.
Addressing MPs on Tuesday, for the first time since he was fined by Met Police for breaking Covid laws, Mr Johnson apologised for his “mistake” 35 times.
He said he had not realised he was breaking the rules when he attended a birthday gathering in Downing Street but that he accepted the police’s decisions.
Labour leader Sir Keir Starmer accused the prime minister of offering “absurd” excuses; while the SNP’s Westminster leader Ian Blackford said: “If he has any decency, any dignity, he would not just apologise, he would resign.”
And Liberal Democrat leader Sir Ed Davey said it was “profoundly damaging” to the UK to be “led by a man the public no longer trust and no longer have confidence in”.
In the Commons, the majority of Conservative MPs gave the PM their backing, but a former senior minister – Mark Harper – urged Mr Johnson to quit saying his actions had been “indefensible”.
And another MP Duncan Baker told Radio Norfolk: “If there are more fixed penalty notices MPs like me will have a lot to consider.”
After taking questions in the Commons for over an hour and a half, Mr Johnson address his own MPs in a bid to shore up support.
The BBC has been told Mr Johnson asked his party if they would rather have him or Labour in power, adding: “We’re going to get on with our one-nation Conservative agenda.”
A Downing Street source has confirmed that Mr Johnson told Conservative MPs the government’s plan to send some asylum seekers to Rwanda had been “misconstrued” by the Archbishop of Canterbury and the BBC.
The Archbishop of Canterbury, Justin Welby, said the policy was “the opposite of the nature of God” during his Easter sermon at the weekend.
Mr Scully told the Today programme that he did not remember the PM’s remarks, and said: “I am not attacking the BBC.”
Mr Johnson will face MPs again at midday when he goes to the Commons for Prime Minister’s Questions.
What has the PM told MPs about parties?
When asked whether there was a party in Downing Street on 18 December 2020, Boris Johnson told the Commons on 1 December 2021 that “all guidance was followed completely in No 10”.
After the publication of a video showing No 10 staff joking about the 18 December event, he told MPs on 8 December 2021 he had been “repeatedly assured” that “there was no party and that no Covid rules were broken”.
Later that day, he told the Commons he was “sure that whatever happened, the guidance was followed and the rules were followed at all times”.
On 12 January 2022, he apologised for attending a Downing Street garden party on 20 May 2020 but said he had “believed implicitly” it was a work event.