The story, carried by the Zambian Observer, said that Swazi men would have to marry several wives, starting from June.
The story, first published on Monday, said the government would sponsor marriage ceremonies and offer houses to men who entered into polygamous marriages.
It said the king “has declared in Swaziland that men will from June 2019 be required to marry at least two or more wives, or be jailed if they fail to do so”.
Government spokesman Percy Simelane called the story “malicious” and “poisonous”.
“His Majesty has not made any pronouncement to that effect, as it has never been an issue raised by the people,” Simelane said.
He said the story was “not only an insult to the monarchy and the culture of eSwatini but a disgrace to journalism”.
The government has demanded that the newspaper retract its story.
The article caused a frenzy on Twitter, which had South African men supposedly wanting to cross borders to eSwatini.
@Gideon73792687 tweeted: “News coming in from Eswatini not Swaziland is that the king denies these reports and labelled them as poisonous reports. But I’m still coming though King.”
Another Twitter user @ncholokabelo claimed to have been personally invited.
“I got an invite by Swaziland to open an office there. I told my wife and that I will need to abide by the country laws to avoid imprisonment,” he wrote.
@Map_Welsh appeared to be disappointed that the news may be fake.
“Please don’t hurt our feelings. I have divorced my wife and left Botswana for Eswatini. I need at least five beautiful Eswatini women,” he tweeted.
Despite the eSwatini government demanding the Zambian Observer to retract its story, more tweets continued flooding in under hashtags like #Swaziland, #eSwatini and #KingMswati.