The Prime Minister asked new ethics adviser Sir Laurie Magnus on Monday to assess whether the former chancellor breached the ministerial code with the HMRC settlement.
Mr Sunak said he was not leader at the time and insisted the advice he received when he gave Mr Zahawi a Cabinet-attending role was that there was “no reason” not to appoint him.
He defied Labour demands to sack the minister, who he insisted “acted properly throughout” amid questions as he welcomed the chance to answer “any and all specific questions in a formal setting” amid concerns about the settlement estimated at around £5 million.
Mr Sunak said he was launching an investigation because “clearly in this case there are questions that need answering”.
“That’s why I’ve asked our independent adviser to get to the bottom of everything, to investigate the matter fully and establish all the facts and provide advice to me on Nadhim Zahawi’s compliance with the ministerial code,” he told broadcasters during a visit to a Northampton hospital.
“I’m pleased that Nadhim Zahawi has agreed with that approach and has agreed to fully co-operate with that investigation.”
Mr Zahawi was understood to have paid a penalty – reported by the Guardian to be around 30 per cent – during the time he was chancellor under Boris Johnson between July and September.
It was one of two scandals Mr Sunak was dealing with relating to Mr Johnson’s period in No 10, with BBC chairman Richard Sharp reportedly helping the then-prime minister secure a loan of up to £800,000.
Mr Sunak distanced himself from the allegations surrounding Mr Zahawi, who he appointed chairman in October.
“The questions that are being asked relate to a time before I was Prime Minister. When I was Prime Minister, the advice that I received was that there was no reason why Nadhim Zahawi could not be appointed to government,” he said.
Mr Sunak insisted it was “longstanding practice” for Mr Zahawi to continue in the role while under investigation.
In a statement, Mr Zahawi said: “I welcome the Prime Minister’s referral of this matter to the independent adviser on ministerial standards. I look forward to explaining the facts of this issue to Sir Laurie Magnus and his team.
“I am confident I acted properly throughout and look forward to answering any and all specific questions in a formal setting to Sir Laurie.”
He said it would be “inappropriate to discuss this issue any further” while continuing as Conservative chairman.
One ally had been adamant Mr Zahawi would be staying on, saying: “He is absolutely not resigning.”
Mr Zahawi, the MP for Stratford-upon-Avon, has insisted his “error” over shares in the YouGov polling company he co-founded was “careless and not deliberate”.
He was also facing new allegations that he falsely told officials he had not exchanged WhatsApp messages with Conservative former prime minister David Cameron, who was lobbying for government loans for Greensill Capital.
The Times reported that it later emerged the pair had discussed the since-collapsed firm when messages were released to a select committee inquiry.
On Monday, Mr Zahawi said “morning” to waiting reporters and cameras as he arrived at Conservative Campaign Headquarters (CCHQ), but gave no comment on his financial affairs or position.
Former Downing Street communications chief Sir Craig Oliver said Mr Zahawi is “hanging on by a thread”.
“I think he’s in serious trouble, you cannot be Conservative Party chairman and not go out and face the media,” he told BBC Radio 4’s Today programme.
“The problem for Nadhim Zahawi at the moment is it doesn’t all add up. Why did you take the job as chancellor when you were clearly in dispute with the HMRC, and he is yet to come out with an answer that is satisfying or feels comfortable on that point.”
Speaking to journalists at a press conference in Edinburgh, Nicola Sturgeon called on Mr Zahawi to quit.
She said: “I think his position … is untenable. I think he should resign his current position as chair of the Conservative Party.
“It seems to be the case … that while he was Chancellor he settled a seven-figure tax bill with HMRC which included a very significant penalty.
“That means, as far as I can see, that there must have been something untoward about his tax affairs to a very large tune, in terms of the amount of money involved.
“I think somebody who is a government minister in that position, I just think their position is untenable.
“I would think he should resign his post, and if he doesn’t do so – rather than kicking it into the long grass and instructing an inquiry as the Prime Minister has done this morning – I think the Prime Minister should remove him from office.”
A challenging weekend for Mr Sunak began when police fined him for failing to wear a seatbelt as he filmed a social media clip in the back of a moving car.
On Monday, he told broadcasters: “In this instance, I made a mistake which I regret deeply and that’s why I apologised straight away.”
It was his second police fine – after paying a fixed-penalty notice during the partygate affair and strained Mr Sunak’s vow on entering No 10 to ensure “integrity, professionalism and accountability at every level of the Government”.