Prime Minister David Cameron has declared his "full support" for Culture Secretary Maria Miller after the parliamentary sleaze watchdog opened an investigation into her expenses.

The office of John Lyon, the Parliamentary Commissioner for Standards, said he was opening an inquiry following a complaint that Mrs Miller had claimed more than £90,000 in second home allowances towards the cost of a house where her parents lived. Mrs Miller has insisted that her expenses were "absolutely as they should be".

But the case has ignited hostilities between Downing Street and the press, after Mr Cameron's director of communications Craig Oliver phoned The Daily Telegraph's editor about the story. Mr Oliver reportedly reminded editor Tony Gallagher during the call that Mrs Miller was in charge of the Government's response to the Leveson report on press regulation, though Number 10 insisted that he was merely highlighting concerns about the way reporters conducted their investigation into her expenses, rather than attempting to threaten the paper.

Arriving in Brussels for a European Council summit, Mr Cameron said: "Maria Miller does an absolutely excellent job as Culture Secretary and she has my full support. A newspaper has asked her a number of questions. So far as I can see, she has got excellent answers to all those questions. I am sure she will answer them and then get on with her job, which is what she should do."

Mrs Miller claimed second home allowances of £90,718 - almost the maximum permitted - between 2005 and 2009 towards mortgage payments, bills and other costs relating to a house where her parents had apparently been living since 1996.

Labour MP John Mann complained to Mr Lyon earlier this week about the arrangements, which he said were "identical" to those of former Labour minister Tony McNulty, who in 2009 was required to pay back more than £13,000 in expenses claimed on a second home occupied by his parents. In that case, the Commissioner said Mr McNulty had effectively "subsidised" his parents from the public purse by allowing them to live rent-free. But a spokesman for the Culture Secretary said: "Mrs Miller's expenses have been audited twice and found to be wholly proper and above board. Any suggestion to the contrary is simply untrue." The Basingstoke MP was ready to "fully co-operate" with any inquiry, said the spokesman.

In an interview with the Evening Standard, Mrs Miller said one of the two audits was carried out by former civil servant Sir Thomas Legg - who was called in to review all MPs' claims at the height of the expenses scandal - and the other by the Conservative Party. Asked whether Sir Thomas was aware that her parents were living at her designated second home, Mrs Miller said: "I obviously spoke to the Fees Office about my claims and they were happy that everything was in order."

She also denied she had used her position overseeing the post-Leveson reforms of press regulation to attempt to ward off inquiries from the Telegraph. After the Telegraph first raised the issue of Mrs Miller's expenses last week, her aide Joanna Hindley is reported to have called the paper to voice concerns about the Cabinet minister's elderly father being "doorstepped" at his home. During the conversation, the aide reportedly said she wanted to "flag up" the fact that the Culture Secretary had been talking with Fleet Street editors about the Leveson Report, which recommended a new structure for press regulation. It later emerged that Mr Oliver had called Mr Gallagher, reportedly telling him "she (Maria Miller) is looking at Leveson at the moment".

But Mrs Miller told the Standard: "This has nothing to do with the Leveson inquiry. My concern is that any investigation is done in accordance with the rules, the Editors' Code. What I did was to contact the editor of the Telegraph directly to express my concern at the way his investigation was being undertaken." She added: "The journalist hadn't contacted my office first. She had doorstepped a member of my family, a person who is not in public life, a person ill-equipped to deal with national media inquiries on my behalf."

No 10 refused to confirm whether the Prime Minister and Culture Secretary had met or talked by telephone about the expense claims. Officials also refused to say if Mr Cameron was relying on Mrs Miller's account of the claims or if he had taken any further action to investigate the allegations.