Kainerugaba, 52, has long been touted as a possible successor to his ageing father and is well known for his fiery social media posts including threats to behead the leading opposition leader Bobi Wine.
He played a key role in securing his father's re-election in the last polls in January by ordering a crackdown on opponents and cutting the internet.
"I will stand for the highest office in the land in 2031. And we will win by the grace of Almighty God," Kainerugaba said in one of a series of posts on the X platform late on Tuesday.
"With my great father, Mzee's, support I will get 90% in the elections in 2031," he added using an honorific reference to his 81-year-old father.
Kainerugaba, who leads a political pressure group called the Patriotic League of Uganda, had previously declared he would stand in the last election in January but ultimately backed out.
Museveni, who heads the ruling National Resistance Movement, has ruled Uganda since 1986 and critics have accused his government of a range of human rights violations including torture, kidnappings and arbitrary detentions. The government denies the allegations.
The country's main opposition leader, Bobi Wine, fled to the U.S. earlier this year, saying he feared for his life. Another veteran opposition figure, Kizza Besigye, has been in prison since 2024 and is on trial for treason.
When asked if Museveni would seek another term, government spokesman Alan Kasujja told Reuters, "as has been the case over the years, President Museveni and the party he leads will make that decision when the time comes."
(Reporting by Nairobi Newsroom; Editing by Ammu Kannampilly and Andrew Heavens)

A campaign billboard of President Yoweri Museveni (C), is seen outside the Entebbe International Airport, in Entebbe, Uganda on 13 January 2026 (issued 14 January 2026). Eight presidential candidates, including Museveni and NUP candidate Bobi Wine, are vying for the presidency in Uganda, scheduled for 15 January. Museveni seeks a seventh term after four decades in power. EPA/Daniel Irungu
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