The announcement, made in a social media post, says it would be made at the Hanwha Philly Shipyard, which South Korea bought to boost its business with the U.S. Navy.
The permission comes after Trump and President Lee Jae Myung reached a trade agreement during a bilateral meeting on Wednesday.
President Trump is breaking with tradition in agreeing to share highly sensitive technology with South Korea.
The cost is unclear.
Trump said during the meeting several economic promises were made on the part of South Korea.
"Our Military Alliance is stronger than ever before and, based on that, I have given them approval to build a Nuclear Powered Submarine, rather than the old fashioned, and far less nimble, diesel powered Submarines that they have now. A great trip, with a great Prime Minister!" Trump posted on his social media platform on Wednesday.
The economic agreements that Trump said were made during the meeting Wednesday include South Korea paying the U.S. $350 billion in exchange for the Trump administration granting the country a lower tariff rate.
Trump also said that South Korea agreed to buy U.S. oil and gas in "vast quantities," with those investments that will "exceed" $600 billion.
After the meeting, President Lee's chief of staff said the tariffs the U.S. imposes on automobile exports will be lowered to 15% from 25%. The framework deal from July lowered the reciprocal rate to 15% and that will be maintained.
Back in July, South Korea pledged to invest $350 billion into the U.S. Lee's chief of staff said a $200 billion investment will be made in installments of up to $20 billion cash payments each year. Another $150 billion will be invested in the U.S. shipbuilding industry.
This now puts the auto tariffs South Korea will pay in line with those on Japan.
The trade deal also includes a commitment from Korean Air to purchase 103 new Boeing aircraft for $36.2 billion, a move that is expected to support up to 135,000 jobs in the U.S. The Republic of Korea Air Force will invest $2.3 billion to develop its aircraft with an American technology company.
The White House has not responded to ABC News' request for comment about the tariffs.
Earlier on Wednesday, the South Korean president greeted Trump with flattery and gifts, including a replica of the ancient gold crown from the Silla dynasty. Their meeting was held in Gyeongju, South Korea, which was the capital of the ancient Silla Kingdom.
Trump was also awarded the Grand Order of Mugunghwa, South Korea's highest honor. Trump is the first U.S. president to receive the honor.
ABC News contributed to this post.