The McDonnell Douglas DC-9 twin-engine jet, which arrived in the poverty-stricken kingdom on Tuesday, is conservatively estimated to be worth at least R360-million.
Labour movements and banned opposition parties in Swaziland are outraged over the jet, which is to be used by Mswati and his 13 wives.
But the Swazi government has denied buying the aircraft. Instead, a government spokesman said it was a gift from "development partners and friends of the king" who wished to remain anonymous.
The South Africa-based Swaziland Solidarity Network (SNN) accused the government of lying, and insisted taxpayers' money was used to buy it.
"The government hopes to con the world," said Lucky Lukhele, spokesman for the SNN.
He asked how an alleged "development partner remarkably overlooked all the basic developmental needs that the country faces and decided to buy the flying metal for one of the richest men in Africa".
Forbes magazine ranks Mswati as the world's 15th-richest monarch.
The SNN said Swaziland was in the middle of its worst economic crisis ever, with seven out of every 10 of Mswati's estimated 1.1-million subjects living below the international poverty line of $1 a day.
But the king's nephew was adamant that the private jet was from Kuwait.
"His Majesty did not buy the jet ... [it] was a gift from Kuwait," Dlamini, 21, said.
He would not elaborate, and Mswati's press office has declined to comment.
According to London's Daily Telegraph newspaper, Swaziland's critics have asked who the "development" partners might be and the reason for their generosity.
The paper said Qatar and Kuwait had close ties to the kingdom, and are believed to have helped finance a £300-million international airport outside the capital.
Mswati, who has ruled Swaziland since 1986, turned 44 on April 19, celebrating with a R3-million party paid for, in part, by the state.