The suspect, identified only as Hussein, was seized as he tried to sneak into Kenya at the Busia border crossing, police spokesperson Judith Nabakooba said.
However, Ms Nabakooba could not provide details about the suspect, saying she was yet to get briefing from the Counter Terrorism Police head, Mr John Ndingutse.
By press time it could not be established whether Hussein is linked to the attack in Nairobi last week in which one person was killed and 35 others injured.
On Sunday, the Uganda Police mounted roadblocks in Mpigi district, 30 kilometres east of Kampala after a tip-off that four suspicious men in a dark grey vehicle were in the area. Police failed to track the vehicle.
Ms Nabakooba said on Monday that the Joint Intelligence Committee (JIC) that includes operatives from the Ugandan military intelligence, police internal and external security organisations were jointly hunting for three other terror suspects who sneaked into Uganda from Kenya last week.
A photograph of one of the suspects was released. The men are believed to be linked to Al-Qaeda but are in the region to reinforce Al-Shabaab, the Somali militant group.
One of the suspects, Ahmed Khaled alias Andreas Martin Muller, according to the Uganda counterterrorism police, is suspected to be travelling with his 33-year-old wife, Kriya Mohamed Hafos Adem, a Somali born in Eritrea and their six-year-old daughter.
The other is reportedly travelling under the names Emrah Erdogan alias Imraan Al-Kurdy Alias Salahaddin Al-Kurdy.
Whereas Muller is German, Al-Kurdy, 24, is a German of Turkish origin.
Last week, the Uganda police chief, Lieutenant General Kale Kayihura, said another terrorism suspect had sneaked into Uganda on a bus from Nairobi.
Source: Daily Nation