The inquiry has rejected the culture secretary's request to bring forward his appearance - expected in mid-May.
Mr Hunt says his evidence will show he acted with "total integrity" but Labour and some Lib Dems want a new probe.
No 10 said the PM "would of course act" if Mr Hunt's evidence suggested he had breached the ministerial code.
The culture secretary has been under pressure since documents released to the inquiry revealed close contact between his special adviser and News Corporation during its planned takeover of satellite broadcaster BSkyB.
Mr Hunt was meant to be acting in what he himself described as a "quasi-judicial" role in deciding whether the proposed merger should be referred to the Competition Commission for final approval.
His special adviser Adam Smith has resigned over what he admitted was an inappropriately close relationship with News Corporation.
The minister has said he did not know about the extent of the contact between the media giant and Mr Smith.
But under the ministerial code of conduct, Mr Hunt is responsible for the actions of his special advisers.
The Downing Street statement was issued this morning as Liberal Democrats lined up to insist that Mr Hunt should face an inquiry under the code.
The Culture Secretary faces allegations that his special adviser disclosed confidential details of the minister’s “quasi-judicial” decisions about the takeover bid to Rupert Murdoch’s News Corp.
Downing Street said today that no inquiry could be launched before the Leveson appearance Mr Hunt will make next month at the earliest, but significantly did not rule it out after that hearing.
"We have always been clear that the Prime Minister and not the Leveson Inquiry is the arbiter of the ministerial code," a spokesman said.
"Jeremy Hunt will be appearing before the inquiry under oath and has made clear he will be providing all necessary evidence for consideration.
"It does not make sense to cut across a judicial inquiry with a parallel process that would risk pre-empting, duplicating or contradicting it.
"Once Jeremy Hunt's evidence is made public and he is questioned, if there is anything that suggests there has been a breach of the code the prime minister would of course act."
The deputy chairman of the Conservative Party, Michael Fallon, also said that the evidence should be "tested" at the Leveson Inquiry - but be subject, if necessary, to a Whitehall inquiry afterwards.
"If later on, obviously, it looks as if there's something that needs to be investigated under the ministerial code that can be done," he said.
The Culture Secretary is due to give evidence, including private text messages and emails, to the inquiry within weeks, but the judge is not expected to report until October at the earliest. No date has yet been set for Mr Hunt’s appearance, but politicians will not be called until mid-May.
A growing number of senior figures suggested that Sir Alex Allan, the independent adviser on the Ministerial Code, should be asked to investigate the specific allegations against the minister.
Sir Christopher Kelly, chairman of the committee on standards in public life, called for the allegations to be “properly investigated” and “reasonably quickly”.
He said: “One obvious way to do it is by asking the Independent Adviser on Ministerial Interests [Sir Alex] to look at them.” Alternatively if Lord Justice Leveson was to rule on the code, that role should be put “beyond doubt”.
Nick Clegg, the Deputy Prime Minister, backed Mr Cameron but also suggested that a Ministerial Code investigation could follow Mr Hunt’s appearance before the Leveson Inquiry.