James Cleverly said it was the job of Parliament to hold the Prime Minister to account, not the London Assembly.
Members of the Assembly on Tuesday gave the PM a two-week deadline to provide "details and a timeline of all contact with Jennifer Arcuri including social, personal and professional during his period of office as mayor of London".
Refusing to comply could see the Conservative leader formally summoned by City Hall and in breach of the law.
The Sunday Times reported that Ms Arcuri, an American who moved to London seven years ago, was given £126,000 in public money and was treated to privileged access to three foreign trade missions led by Mr Johnson while he was mayor of London.
Mr Cleverly, who previously served on the London Assembly, poured doubt on whether the PM would comply with the Assembly's demands for information.
"The London Assembly's job is to scrutinise the mayor of London, and in that capacity they need to focus on the failings and ineptitude of the current mayor," he told BBC Radio 4's Today programme.
"When I was on the London Assembly, it was quite common for Government ministers to refuse to appear.
"The body that exists to scrutinise ministers is the House of Commons, the body that exists to scrutinise the mayor of London is the London Assembly, and there is a clear division and role there."
When told the PM could be in breach of the law for failing to explain himself, Mr Cleverly replied: "That's not my understanding. I'm willing to look into it."
The Government has frozen a £100,000 grant to Ms Arcuri's company, Hacker House, and is facing embarrassing questions about the verification process carried about before awarding the money.
Digital Minister Matt Warman said his department had done the "usual due diligence" and that the company had a British phone number.
Yet numerous reports said calls to the number were directed to an office in California, where Ms Arcuri is said to now be based.
Mr Warman said: "This is a company that is based in Britain as far as Companies House is concerned. It is a company with a British phone number.
"We will review that, but we have no reason to think that there is anything untoward in this particular matter."
Almost half the money awarded has already been handed to the training outfit.
Asked previously about the allegations, Mr Johnson said "everything was done with complete propriety" but declined to clarify the nature of his relationship with Ms Arcuri, 34.