No 10 said: "We don't accept threats have been made."
It is understood spokesman Craig Oliver phoned the Daily Telegraph's editor after his paper wrote a story about Maria Miller's expense claims.
He reportedly told him the story was poorly timed.
Mr Oliver is alleged, by the Daily Telegraph, to have told the editor, Tony Gallagher, that Mrs Miller was "looking at Leveson at the moment".
Twenty-four hours earlier, Mrs Miller's special adviser Joanna Hindley called the  reporter working on the story to "flag up" the culture secretary's role in drawing up new press regulation rules, following Lord Justice Leveson's inquiry into media ethics.
A No 10 spokesman defended Mr Oliver's intervention and rejected suggestions he - or Mrs Miller's  adviser - were seeking to influence the paper against pursuing the story.
They said Mr Oliver was simply raising "concerns" with the newspaper about the way the story was being pursued and the fact Mrs Miller's "elderly father had been door-stepped" by a reporter.
The spokesman added: "The secretary of state had some concerns about the way that investigation was being conducted and Craig Oliver was simply reflecting those concerns."
Labour MP John Mann has written to John Lyon, Parliamentary Commissioner for Standards, after it emerged Mrs Miller had allowed her parents to live at a property on which she claimed Â£90,718 in second home allowances during the last parliament.
Mrs Miller has said her expenses were "absolutely in order" and "in complete accordance with the rules".
Her parents, John and June Lewis, have apparently been living at the property since selling their home in Wales in 1996.
According to the Telegraph, Miss Hindley told its reporter: "Maria has obviously been having quite a lot of editors' meetings around Leveson at the moment. So I am just going to kind of flag up that connection for you to think about."
Brian Cathcart, executive director of Hacked Off, which is campaigning for the setting up of an independent press regulator via a change in the law, said: "This story illustrates exactly why ministers must be kept at arm's length from the regulation of the press.
"It cannot be right that politicians who are subject to the scrutiny of the newspapers and who are constantly vulnerable to public challenge in this way are sitting down with editors and proprietors of those same newspapers to design a press regulation system."
A Department for Culture, Media and Sport spokesman said: "Mrs Miller's special adviser raised concerns with a journalist about the nature of an approach to Mrs Miller's elderly father. Her adviser noted that Mrs Miller was in regular contact with the paper's editor and would raise her concerns directly with him, which Mrs Miller did subsequently.
"However, this is a separate issue to ongoing discussions about press regulation. Mrs Miller has made the government's position on this clear."
Downing Street has denied that the PM's spokesman warned a newspaper against running a critical story on the culture secretary's expenses because of her role in enacting the Leveson proposals.