Peter Robinson has asked for an inquiry into the affair
Northern Ireland's first minister should be given the chance to clear his name, the secretary of state has said.
Shaun Woodward said Peter Robinson "fiercely resisted" allegations of financial wrongdoing and had asked officials to investigate the matter.
"I think we have to allow that to happen," Mr Woodward said.
A BBC programme has said Mr Robinson did not tell the authorities his wife Iris had not registered £50,000 she obtained from two property developers.
The Spotlight programme said she obtained the money from two property developers, which was paid to her 19-year-old lover to help him launch a cafe.
It said DUP leader Mr Robinson knew about her financial dealings but did not tell the proper authorities despite being being obliged by the ministerial code to act in the public interest at all times.
'Issue of trust'
The financial allegations follow a public admission by Mr Robinson that his wife had attempted suicide after her affair.
Mr Woodward said the matter had been a "personal crisis and a personal tragedy" for the Robinson family, but politicians were entitled to a private life.
The secretary of state also said Mr Robinson had been "a pivotal figure" in the process of devolution in Northern Ireland.
Asked whether public faith in him may have been irreparably damaged, he said: "Trust is of course the essential commodity - whether it's in a political process or indeed in building the original peace process.
"There are no doubts about this that during this extraordinary turn of events the issue of trust does get placed out there into the fore.
"But I think we have all got to be conscious that the [devolution] process is bigger than any one man."
Earlier, senior DUP member Gregory Campbell said his party leader should be given a week to prove that he did not breach any rules.
He added: "Peter himself has asked for a week to resolve those issues, respond to them and refute them."
However, Reverend David McIlveen, a Free Presbyterian minister who is a close friend of former DUP leader Ian Paisley, said he believed Mr Robinson should consider stepping down temporarily.
"I do believe that his position is becoming increasingly untenable," he said.
"He has a major problem with regard to solving his own family difficulties, and I personally cannot take the view that a person's private life does not affect their public life.
"Judgements that we make in private will undoubtedly influence our judgements in public."
In an interview on Friday, Mr Robinson said he had asked officials at the Office of the First and Deputy First Minister to set up an investigation into his conduct.
"I don't believe that I have done anything wrong. I have acted properly at all times. I have subjected myself to investigation. I am prepared publicly to allow the outcome of that investigation to be known," he said.
"I am prepared to act in accordance with the outcome of that investigation and I believe that the public, who are fair in these matters, will be prepared to have me judged by a thorough investigation and legal opinion being given to us, rather than the smear and innuendo of the press."
However he added that he would "not be slow" in taking a decision if the investigation found that he should have acted differently.