Mr Mbabazi made the announcement on You Tube, saying he wanted to breathe “new life into our system of government”.
President Museveni has been in power since 1986 and is expected to seek a fourth term in next year’s poll.
He sacked Mr Mbabazi last year, which some saw as a move to get rid of a potential rival.
Mr Mbabazi said he will first seek the nomination to become the candidate for the governing National Resistance Movement (NRM).
President Museveni has already been endorsed by the party’s national executive committee but that needs to be approved by a party congress.
It is at that meeting, expected later this year, that Mr Mbabazi will formally challenge the president.
BBC Focus on Africa editor Rachael Akidi says that Mr Museveni has never faced a credible challenge within the party.
The former prime minister is thought to have some sway amongst ordinary members as he served as the NRM’s secretary general and was one of the key strategists behind Mr Museveni’s long rule, the BBC’s Catherine Byaruhanga in Kampala says.