Tension is high in the area with the presence of the foreign troops, who are pursuing members of the outlawed Oromo Liberation Front (OLF).
Ethiopian soldiers and OLF fighters had been involved in clashes before the latter escaped by crossing into Kenya.
Acting county police commander Mark Wanjala said the police officers were ambushed while on patrol in the area.
"I can confirm the soldiers have killed one of their own chiefs (at) the border on Thursday. Three police officers were also killed today (Friday)," Mr Wanjala said
Police on Friday engaged the fighters in a fierce gun battle in the area.
Sololo AP Commander Ali Boru, who spoke to the Nation at about 4pm, said police were in the bush engaging the infiltrators.
An Ethiopian chief identified as Guyo Harake is said to have been shot dead on Thursday at the border as soldiers engaged OLF fighters approximately eight kilometers from the Sololo police station.
According to residents, the soldiers engaged the rebels in a gun battle in Anona at around 5pm on Thursday.
Ethiopian soldiers have crossed into Kenya in Moyale and Sololo five times between March and November.
Kenya police reservists lost two government-issued rifles and ammunition to the foreign soldiers.
The Kenyan government has already protested to Ethiopia over the invasions, which former county commissioner Peter Thuku said are unwarranted.
Source : Daily Nation