Ahmed Eid said such access could be granted "soon", raising the possibility that 15 percent of the under-construction King Abdullah stadium in Jeddah could be made into family boxes, where women could watch matches.
The stadium is due to be finished next year.
But the subject immediately triggered a wide-ranging debate in the Saudi media, with unfavourable opinion forcing the official to issue a clarification that he was merely expressing a personal opinion.
Ultra-conservative Saudi Arabia, which applies a strict interpretation of Islamic or Sharia law, bans men and women mixing in public and access for women in public areas.
Women's sport in the kingdom is virtually non-existent, although some young girls do take part in private schools under certain conditions.
"The question is the responsibility of the relevant authorities," said Eid. "The decision to allow women into stadiums is not one for any of the sporting federations."
The head of Saudi sport, Prince Nawaf bin Faisal, said only that he had nothing to add to Eid's clarification.
Saudi Arabi recently saw the first woman in a sporting arena when she and one of her "legal guardians" were at a show-jumping competition in the eastern province of Al-Ahsa.
A "legal guardian" is a Saudi woman's close relative without which she cannot travel overseas or to a public place where men may be present.
Some Saudi women supporters have also travelled to Kuwait to cheer on their football club, Al-Fath, in a regional competition.
But they were turned down when they asked to be able to watch the return leg at home.
In January, South Korean and Japanese expatriate women were allowed to watch matches in a Asian handball competition and female Saudi journalists covered the event.
Saudi Arabia is hoping to organise the finals of football's Asian Cup in 2019.
Should they be successful, they will be required to reserve spaces for women supporters, according to the rules of organisers, the Asian Football Confederation.
The debate comes after a young Dubai-based Saudi woman, Raha Muharraq, on Saturday became the first from the kingdom to scale the world's highest peak, Mount Everest.
Last year, Saudi Arabia came under international pressure to allow two female athletes to compete in the Olympic Games in London.
The International Olympic Committee allowed a judo player and a runner to take part with their heads and body covered, in compliance with Sharia law.