Shota Utiashvili, the Georgian interior ministry spokesman, said that three Georgian police officers had been shot and injured on a patrol mission.
But Garik Kupalba, deputy defence minister in Abkhazia's separtist administration, denied the Georgian account saying the only people injured were two Abkhaz men.
"Two Abkhaz soldiers were injured today ... when their post came under fire" in the Kodori Gorge, which is partly controlled by separatist forces and partly by Georgian forces, he said.
He said they were being treated in hospital.
Airspace row
Meanwhile, Russia and Georgia traded accusations over a number of military jets which flew over South Ossetia, another breakaway region.
A spokesman for Russian peacekeepers based in the region said Georgia flew two fighter jets over South Ossetia on Tuesday, but Georgia said the military planes were in fact Russian and that there were four of them.
Russia's foreign ministry had earlier accused Georgia of "multiple breaches of airspace in the conflict zone by fighter jets and spy planes."
But Colonel Zurab Pochkua, the deputy commander of Georgia's air force, said: "On July 8 at 8:11 pm [16:11 GMT], two Russian fighter jets and at 8:20 pm two more aircraft entered Georgian airspace.
"For almost 40 minutes, four Russian planes were circling over the territory north of Tskhinvali," the main city in South Ossetia, he said.
A flight ban is in place over South Ossetia as part of ceasefire arrangements in the area.
Rice visit
Rice's trip arrives in Georgia after signing a deal allowing Washington to base part of a new US missile defence system in the Czech Republic.
The move prompted a stark warning from Moscow that it would respond militarily.
Washington and Moscow have also traded accusations of fanning tensions in Georgia itself, where violence has flared in the past week in Abkhazia, one of two regions that broke from Tbilisi after the collapse of the Soviet Union.
Rice is scheduled to attend a private dinner with Mikheil Saakashvili, Georgia's US-backed president, on Wednesday evening.
Hours before her arrival, the Russian foreign ministry issued a statement accusing Georgia of pushing the region towards war through actions openly supported by the United States.
"Tbilisi's actions represent a real threat to peace and security in the south caucasus, capable of taking the region to the brink of a new armed conflict with unpredictable consequences," the ministry said in a statement.
Abkhazia and South Ossetia broke away from the rest of Georgia during conflicts in the early 1990s that killed several thousand people.