The prime minister was informed of Brooks's decision to quit at Chequers today shortly before a meeting of the cabinet. His spokesman said: "He thinks it's the right decision. He said the other day he would have accepted her resignation."
Asked whether Brooks should still give evidence to the culture, media and sport select committee, the spokesman added: "I don't think [Cameron] would change that. I presume the committee will still want to see her."
Brooks, Rupert Murdoch and James Murdoch have agreed to appear before the culture committee next Tuesday to answer questions about phone hacking. In a statement today, News International chairman James Murdoch said that the session would give them a chance to show their "determination to put things right".
Discussing Brooks's resignation, a spokesman for deputy prime minister Nick Clegg said: "This is the right thing for Rebekah Brooks to have done. It is an important first step in cleaning up this mess.
"People will, rightly, expect Brooks to come to the select committee next week to give evidence. People still need answers. She owes it to the victims of phone hacking and the country at large to explain her role in what happened."
Rebekah Brooks was editor of the News of the World between 2000 and 2003, during which time the newspaper hacked the phone of murdered schoolgirl Milly Dowler.
Announcing her resignation today, Brooks said that she felt "a deep sense of responsibility" for the people who have been hurt by the phone hacking scandal, and said that she wanted to "reiterate how sorry I am for what we now know to have taken place".
> Read her resignation statement in full
Labour leader Ed Miliband, who originally called for Brooks to quit on July 5, also welcomed the resignation, but said that News Corp chairman Rupert Murdoch still has many questions to answer.
"It is right that Rebekah Brooks has finally taken responsibility for the terrible events that happened on her watch, like the hacking of Milly Dowler's phone. No one in this country should exercise power without responsibility," he said.
"But as I said when I called for her resignation ten days ago, this is not just about one individual but about the culture of an organisation. Rupert Murdoch says that News Corp has handled these allegations 'extremely well'.
"He still hasn't apologised to the innocent victims of hacking. He clearly still doesn't get it. When he comes to the House of Commons next week, people will expect him to start taking some responsibility and apologise for the illegal actions which happened in his organisation."
Tom Mockridge, the chief executive of Sky Italia, has been named as the chief executive of News International and will start immediately.