Mr Museveni has long been suspected of grooming the 48-year-old to succeed him when he eventually steps down.
President Yoweri Museveni has asked Kenyans for "forgiveness" after his son who was the commander of Uganda’s land forces, tweeted about invading their country.
His apology came a day after he sacked his son, Muhoozi Kainerugaba as commander of the army's land forces, BBC reports.
Kainerugaba was removed from the position after he repeatedly threatened on Twitter to invade neighbouring Kenya.
He was however promoted to the rank of a full general, the highest rank in the army and will retain his position as presidential adviser on special operations.
Kainerugaba, widely regarded as the de facto head of the military and his father's chosen successor, later said the comments were made in jest, Reuters reports.
Mr Museveni has long been suspected of grooming the 48-year-old to succeed him when he eventually steps down.
Ugandan analysts and opposition leaders have long accused the 78-year-old Museveni of grooming his son to take over from him, but Museveni, who has been in power for 36 years, has repeatedly denied doing so.
Kainerugaba is outspoken on social media, frequently trading barbs with opposition figures and weighing into politics, even though his military role bars him from doing so.
On Monday and Tuesday, he sent a series of provocative messages on Twitter, including proposing the unification of Kenya and Uganda and offering cows to marry Italy's likely next leader.
"It wouldn't take us, my army and me, 2 weeks to capture Nairobi," Kainerugaba wrote, referring to Kenya's capital.
"Union is a MUST! No honourable men can allow these artificial, colonial borders anymore. If we our generation has men then these borders must fall!"
His comments attracted public outrage from Kenyans on social media.
It led to an uproar on Twitter, with people accusing Gen Kainerugaba of being "reckless" and jeopardising relations between the neighbouring states.
"I would never beat up the Kenyan army because my father told me never to attempt it! So our people in Kenya should relax!" Gen Kainerugaba tweeted.
He also urged the aggrieved Kenyans to "relax".
He said he had spoken to Kenya's former President Uhuru Kenyatta, and expressed regret that he had stepped down at the end of his two terms in August.
"My only problem with my beloved big brother is that he didn't stand for a third term. We would have won easily!" Gen Kainerugaba said.
Although the Kenyan government did not comment, Uganda's foreign ministry issued a statement saying Uganda treasured its "strong bilateral relations" with it "brotherly neighbour".
In the latest attempt at damage-control, Mr Museveni said he had spoken to Kenya's President William Ruto, and he was "very sorry" about his son's tweets.
"I ask our Kenyan brothers and sisters to forgive us for tweets sent by General Muhoozi, former Commander of Land Forces here, regarding the election matters in that great country.
"It is not correct for Public officers, be they civilian or military, to comment or interfere in any way, in the internal affairs of brother countries," Mr Museveni said.
He added that despite his son's "mistake", he had promoted him to the rank of full general because he could still make "many positive contributions".
"I know for a fact that General Muhoozi is a passionate Pan-Africanist," Mr Museveni said.
On Monday, Kainerugaba also asked his more than 600,000 Twitter followers how many cows should be offered as a bride price for Giorgia Meloni, the right-wing politician expected to be named Italy's prime minister this month.
"I would give her 100 Nkore cows immediately! For being fearless and true!!" he wrote.