"Well, in Britain's history we've had some very intense times of terrorists threats," he said, "so I wouldn't want to try to scale it in that way. But certainly, we face a very severe threat. That's what we're calling it, severe, because we believe an attack is highly likely."
Asked whether the world is at war against terrorism, Cameron replied: "What I don't want to do is try to posit that there's some clash of civilizations going on, because that is what the terrorists want. They want this to be seen as a war between what they see as the true Islam, and the rest, and that's not the case. What these terrorists represent is the perversion of a major religion, and not its true adherence."
He was also asked about Pope Francis's statement earlier this week that "one cannot make fun of faith."
"I'm a Christian," Cameron said. "If someone says something offensive about Jesus, I might find that offensive, but in a free society I don't have a right to wreak my vengeance upon them."
He later added: "My job is not to tell a newspaper what to publish or what not to publish. My job is to uphold the law that they can publish things within the law."
Looking back at his trip this week, Cameron said that the partnership between his country and the United States is built "on the fact that we see the world's challenges in the same way and we want to work together to solve them. I believe it's helped by the fact that I have a very strong relationship with President Obama and I admire his leadership and his reasonable approach in dealing with all these issues."
He returned to that point later in the interview. "We've been thrown together in a lot of different situations," he said. "And we don't always agree, but what I find is he has a very calm and reasonable, rational, sensible, level-headed way of trying to address these great challenges. And it's been a pleasure to work with him."
Host Bob Schieffer then asked a somewhat unexpected question: "You say he calls you 'bro.' What do you call him?"
"Barack," apparently. "But every now and again, there has been some other things thrown around, and I suppose -- 'bro' was one of them -- but I took that in good heart."
Schieffer ended the interview by asking if Cameron had a "special nickname" for the president of the United States.
"I don't," he said. "No. If I did, I probably wouldn't tell you."