The Prime Minister's office said afterwards that it had no comment to make on the matter.
Beckham, who plays for Los Angeles Galaxy after spending most of his career at Manchester United and then Real Madrid, is estimated to have earned more than $300 million from football and endorsements.
He and his wife Victoria, a former Spice Girl, live in Los Angeles with their four children.
Mr Key would not be the first person to question Beckham's intelligence. One of his celebrated quotes is: "I definitely want Brooklyn to be christened, but I don't know into what religion yet."
Later Mr Key played a round of golf with former All Black captain Sir Brian Lochore, former All Black player and coach Laurie Mains and former Otago rugby player Brett McCormack.
The game at the Balmacewen Golf Course was part of a charity event that netted thousands for the Edgar National Centre for Diabetes Research.
Mr Key said he relished the opportunity to play 18 holes, as his busy schedule usually prevented it.