Norway's Crown Princess Mette-Marit has given a "profound apology" to Norwegians for her friendship with late sex offender Jeffrey Epstein, after it emerged they had exchanged messages for three years.

"I also apologise for the situation in which I have placed the Royal House, especially the King and Queen," she said in a palace statement after days of mounting pressure to address the issue.

Harald V and Queen Sonja, who are both 88, have not commented on the revelations.

The King met Prime Minister Jonas Gahr Støre on Friday, alongside Crown Prince Haakon, who told reporters later that his wife wanted to speak: "But right now she can't, and I also tell her she is not allowed to."

Hundreds of emails from 2011-14 involving Epstein and the crown princess came to light in the cache of files published by the US justice department a week ago.