Kenyan police said on Monday the fighting around the Somali town of Bura Hachi erupted between militiamen from the Marahan clan and their rivals from the Garri clan on Saturday.
The Somali factions used heavy artillery, including anti-aircraft weapons, machine guns mounted on pickup trucks and mortars in the clashes, a police officer said.
"Around 14 Marahans and six Garres were killed ... but the number of dead could be much higher," the officer said, adding that it had not been possible to establish whether the dead were civilians or armed men.
Nairobi's Daily Nation newspaper reported that up to 36 people had died in the clashes.
Kenyan authorities said casualities were brought into Kenya for treatment, and expressed fear that possible revenge attacks might spill across the border.
"We have reinforced security on the Kenyan side because we know the fighting may spill over to the Kenyan side and bring new problems," northeastern Police Commander Gabriel Ndolo said.
Bura Hachi, about 3km from the border, has been hotly contested by the two clans, who have been at loggerheads since Somalia was plunged into anarchy after the 1991 ouster of leader Muhammad Siad Barri.