The first parts of the Terminal High Altitude Area Defense (THAAD) system have already been delivered to a former golf course in the South -- infuriating China -- at a time of heightened tensions over Pyongyang's nuclear and missile programmes.
Top US officials have said THAAD will be operational "within days".
"I informed South Korea it would be appropriate if they paid. It's a billion-dollar system," Trump was quoted as saying by the Reuters news agency. "It's phenomenal, shoots missiles right out of the sky."
The two countries have been in a security alliance since the 1950-53 Korean war, and more than 28,000 US troops are stationed in the South.
Seoul retorted that under the Status of Forces Agreement that governs the US military presence in the country, the South would provide the THAAD site and infrastructure while the US would pay to deploy and operate it.
"There is no change to this basic position," South Korea's defence ministry said in a statement.
The row comes with tensions high on the Korean peninsula following a series of missile launches by the North and warnings from the Trump administration that military action was an "option on the table".
Trump said there was "a chance" of "a major, major conflict" with the North -- which would put the South, whose capital is within range of Pyongyang's artillery, at risk of gigantic casualties.
But earlier this week Washington said it would seek stronger sanctions against Pyongyang and held open the possibility of negotiations, with US Pacific Command chief Admiral Harry Harris saying it wanted to bring leader Kim Jong-Un "to his senses, not to his knees".
The White House also wants China to do more to rein in the North, with Trump saying he believed leader Xi Jinping was "trying very hard".
But Beijing has been infuriated by the THAAD deployment, which it fears weaken its own ballistic capabilities and says upsets the regional security balance.
Social media commentators derided Trump's comments. "So he wants to start a war with North Korea and he wants South Korea to pay for it," wrote one Twitter poster.
THAAD is designed to intercept and destroy short and medium-range ballistic missiles during their final phase of flight.
South Koreans are ambivalent over its deployment, with only 51.8 percent in favour in a Korea Research opinion poll last month.
The country has already been hit hard by a series of measures imposed by Beijing as apparent retaliation.
The tourist industry has been hammered by a Chinese ban on tour groups, with visitor numbers from the Asian giant plummeting 40 percent in March.
Lotte Group, a South Korean retail giant, has had to shut down 85 of its 99 stores in China due to boycott calls after the group agreed to provide the golf course where THAAD is being installed to the government. Its accumulated losses as a result are reportedly expected to hit $1 billion in the first half of this year alone.
The Export-Import Bank of Korea said this month that South Korea could suffer up to 16 trillion won ($14 billion) losses from reduced trade and falls in Chinese tourist numbers over the next two years.
In the Reuters interview, Trump also said he would renegotiate or terminate the five-year-old free trade agreement between South Korea and the US.
The issue had raised widespread concerns in Asia's fourth-largest economy when he brought it up on the campaign trail, but appeared to have been put on the back burner in his first 100 days in office.
But Trump said: "It is unacceptable, it is a horrible deal made by Hillary."
(Except for the headline, this story has not been edited by NDTV staff and is published from a syndicated feed.)