The Ugandans were looking to build on a solid third wicket partnership between Joseph Olwenyi and Junior Kwebiiha in their second innings.
Needing 151 runs from 41 overs, they ended the day on 118 for seven, leaving Kenya with an overnight lead of 37 runs. They resume the final day with three wickets on hand to complete the 11 overs remaining.
Earlier Ugandan medium pacer Kenneth Kamukya had the Kenyan batsmen on the ropes when he took four quick wickets within five overs, forcing the hosts to declare their first innings on 307 for six in 89.3 overs.
After being put in to bat first on Friday, Uganda, winners of the first match in the three-nation tournament against Namibia in April were all out for 152 in their first innings.