According to an AFP report, an anonymous official at the ministry said in July, Kim Yong Jin was killed by a firing squad after being accused of being an "anti-party, anti-revolutionary agitator".
According to the officials in Seoul, these are the latest executions and demotions of top North Korean officials since Kim Jong Un assumed power in 2011.
The mass-selling JoongAng Ilbo on Tuesday said two senior officials had been put to death.
If this report turns out to be untrue, Kim Yong-jin may well appear in public or be listed as in attendance at a major public event in Pyongyang.
Kim Yong-chol, who took the helm of the UFD after predecessor Kim Yang-gon was killed in a auto accident in December, was temporarily purged by being sent to a farm in the countryside to undergo ideological reeducation for a high-handed attitude at his job, the official said.
Jeong said Kim was in charge of education affairs, but refused to disclose why and when South Korea believes he was executed.
Late last month, Yonhap cited sources "familiar with the matter" as saying that North Korea publicly executed six officials deemed responsible for the escape of overseas workers to South Korea earlier this year.
The newspaper has also claimed that Hwang Min was executed, "because policy proposals he had pushed for were seen as a direct challenge to the leadership of Kim Jong-un".
The official, who led the country's reconnaissance bureau, is thought to be behind the sinking of a South Korean warship and shelling of a front-line island in 2010.
The spymaster, who was reinstated this month, is likely to be tempted to prove his loyalty by committing provocative acts against the South, the official said.
"Therefore, we are keeping a close tab on the North", he said.
Professor Yang Moo-Jin at the University of North Korean Studies said the vice premier's execution could be indirectly verified when Pyongyang's state media reveals the names of attendees at the government's anniversary ceremony on September 9. Last year, South Korea's spy agency said Kim had his defence chief executed with an anti-aircraft gun for complaining about him and sleeping during a meeting he had presided over.
In April previous year, the country killed then-defense chief Hyon Yong-chol with an anti-aircraft gun because he dozed off during a military event, according to Seoul's spy agency.
The most notorious case was that of Kim's uncle Jang Song Thaek, who was executed for charges including treason and corruption in December 2013.
North Korea's deputy ambassador to Britain sought refuge in the South with his family, the unification ministry said earlier this month.