Omar Bin Laden, 26, and his wife Jane Felix-Browne, 52, say they have been interviewed at the British Embassy in Cairo. The embassy has declined to comment on the issue.
The British woman, who changed her name to Zaina Al Sabah Bin Laden after her marriage to Omar, has been married six times and has three sons and five grandchildren, according to the tabloid.
If the couple's application is accepted, they will move to Jane's $1.1 million home in Cheshire, near Manchester.
The Daily Mail quoted her as saying: "The embassy staff are all very friendly and they are doing all the checks. It could take a while for the visa to come through but there's no reason in law why Omar and I should not be able to live in the U.K. together."
The son of the world's most wanted terrorist has divorced his first wife, the mother of his two-year-old son, and is currently waiting for confirmation of the divorce to come through from Saudi Arabia, so that he can prove the British woman is his only spouse.
"We have been told there will not be a problem as long as we can provide the original documents from his divorce from his first wife. And that should be done in a week," Mrs Bin Laden said.
A British marriage visa would allow Omar to live in the country for two years, after which he would be able to apply for Indefinite Leave to Remain.
The couple say they are "peace activists", and are organizing a horse ride from Cairo to Morocco.
Omar Bin Laden told the Mail on Sunday: "Associates of my father forced the cancellation of the Dakar Rally [across north-west Africa], but they won't stop me from riding. We want people to join us on the trek - Jews, Arabs, Christians, Muslims, it doesn't matter where people are from."
