The meat has long been a part of South Korean cuisine with about one million dogs believed to be eaten annually, but consumption has declined as more people embrace dogs as companions rather than livestock.
The practice is now something of a taboo among younger generations and pressure from animal rights activists has also been mounting.
"Hasn't the time come to prudently consider prohibiting dog meat consumption?" Moon told Prime Minister Kim Boo-kyum during a weekly meeting, according to the presidential spokeswoman.
South Korea's pet industry is on the rise, with a growing number of people living with dogs at home -- the president among them.
Moon is a known dog lover and has several canines at the presidential compound, including a mutt he rescued after taking office.
Adopting Tory was one of Moon's pledges during his presidential campaign and the pooch became the first rescue dog to make its way into the Blue House.
Moon made the remarks as he was briefed on a plan to improve the care system for abandoned pets, his spokeswoman said.
South Korea's current animal protection law is intended mainly to prevent the cruel slaughter of dogs and cats but does not ban consumption itself.
Nonetheless, authorities have invoked the law and other hygiene regulations to crack down on dog farms and restaurants ahead of international events such as the 2018 Pyeongchang Olympics.
