Most Koreans have never eaten dog meat, and demand for it has dwindled in recent years.
But it is estimated that up to one million dogs are still slaughtered for food each year in South Korea.
Speaking during a meeting with the prime minister, Mr Moon questioned whether it was time to "prudently consider" a ban.
It is the first time that the president, a known dog lover, has raised the prospect of a total ban.
He made the comments as he was briefed on new measures to protect abandoned animals in the country.
There is already a law in place banning the cruel slaughter of dogs and cats, but consumption itself is not banned.
However, in recent years, people have turned away from eating dog meat, particularly amid a growing trend to keep the animals as pets.
As a result, three of the country's largest dog meat markets have closed down.
A poll in 2020, conducted by Nielsen for Humane Society International found that 84% of South Koreans have never consumed dog meat or say they do not want to consume it in the future. It also found that 59% of South Koreans support banning dog meat.
Animal rights groups, who have long called for a stop to the dog meat trade, welcomed Mr Moon's comments.
"A growing number of South Koreans are considering the consumption of dog meat as a matter of animal abuse rather than tradition," Jeon Jin-kyung, head of the Korea Animal Rights Advocates, told the Korea Times.
Discussion around the topic is set to intensify in the run up to the presidential election, which is scheduled to take place next year. Several candidates have already raised the possibility of banning dog meat.
Gyeonggi Province Governor Lee Jae-myung, considered a strong candidate, has mentioned it in his campaigning, telling supporters that there need to be new laws and policies based on "social consensus".