Kojori is about 10km from Tbilisi and the military base is home to a special forces battalion.
And at Mount Makhata, about 5km from Tbilisi, a bomb struck an air traffic control centre.
The Georgian government also said Russian aircraft had bombed a military airfield near Tbilisi's main civilian airport on Sunday.
Earlier, Utiashvili said the strategic Georgian town of Gori had come under "massive" attack from Russian artillery and aircraft while tank-led ground forces were preparing for an assault.
"There was massive bombing of Gori all evening and now we are getting reports of an imminent attack by Russian tanks," he said.
"Gori is being bombed massively from the air and from artillery as well," he added.
Temur Yakobashvili, a Georgian interior ministry spokesman, said Russian tanks had tried to cross from South Ossetia into Gori but were turned back by Georgian forces.
Al Jazeera's Jonnah Hull, reporting from Gori, said thousands of Georgians had rapidly evacuated the town at the first mention of a possible full scale Russian invasion.
"There was massive bombing of Gori all evening and now we are getting reports of an imminent attack by Russian tanks"
Georgia interior ministry
But he could not confirm if an invasion was imminent and said there had been no official word from the Georgian government on the attack.
About 65km northwest of the Georgian capital Tbilisi, Gori is just south of the border with South Ossetia, which has seen fierce fighting between Russian and Georgian forces in the last few days.
It is the largest Georgian town, about 50,000 in population, that sits close to the region and an important strategic link between eastern and western Georgia.
Utiashvili said Russian troops were preparing for a ground assault.
They "are not there yet but it looks like they are getting ready for it," he said, adding that Georgian forces were returning fire on Russian positions.
Doubts over ceasefire
The reported attack came after Georgia had offered a limited ceasefire in South Ossetia on Sunday for the passage of humanitarian aid and said it had pulled its troops back across the border.
It also offered immediate talks with Russia for a full ceasefire but did not get an official response.
"Georgia expresses its readiness to immediately start negotiations with the Russian Federation on a ceasefire and termination of hostilities," the Georgian foreign ministry said in a statement on Sunday, adding that it had notified Russia's envoy to Tbilisi.
But Russia said Georgian troops were continuing their attacks and fended off a wave of international calls to observe the Georgian so-called ceasefire, saying it must first be assured that Georgian troops had indeed pulled back from South Ossetia.
Alexander Darchiev, Russia's charge d'affairs in Washington, said Georgian soldiers were "not withdrawing but regrouping, including heavy armour and increased attacks on Tskhinvali".
The US military began flying 2,000 Georgian troops home from Iraq after Georgia recalled them, even while calling for a truce.
Thousands of Georgians gathered in the capital Tbilisi on Sunday to protest against Russia's invasion into South Ossetia, marching to the Russian embassy and United Nations offices and calling for action from the UN and EU to stop Russia.
Also on Sunday, Russia claimed to have sunk a Georgian boat that it said was trying to attack Russian vessels in the Black Sea.
The ITAR-Tass news agency quoted a Russian defence ministry spokesman as saying that Georgian missile boats twice tried to attack Russian ships, which fired back and sank one of the Georgian vessels.
Second front fears
Meanwhile, Georgia said thousands of Russian troops arrived on Sunday in Abkhazia - another breakaway region from Georgia – amid fears of a second front opening as the separatist government there declared its own state of war.
|Russia says at least 2,000 people have been killed in South Ossetia [AFP]
Georgian officials said Russia had begun an operation to storm the Georgian-controlled Kodori gorge in Abkhazia.
But Russia's armed forces on Sunday denied plans to expand their conflict with Georgia into the Abkhazia region.
"We do not plan to escalate the conflict in this region," Anatoly Nogovitsyn, an army spokesman, said in televised remarks, referring to Abkhazia.
In Russia, the number of Ossetian refugees continued to grow.
The Red Cross said the unrest had forced at least 40,000 people from their homes and UN officials said at least 1,000 people had crossed the border into the Russian province of North Ossetia since the war began in South Ossetia.
Three days of fighting have left sections of South Ossetia's capital Tskhinvali in ruins and an undetermined number of fighters and civilians dead.
Grigory Karasin, Russia's deputy foreign minister, said more than 2,000 people had been killed in South Ossetia since Friday, most of them Ossetians, but the figure could not be independently confirmed.
War of words
At the United Nations on Sunday, the Russian and American ambassadors were having a war of their own – exchanging sharp remarks over the escalating conflict.
Zalmay Khalilzad, the US ambassador, accused Moscow of seeking "regime change" in Georgia and resisting attempts to make peace after days of deadly fighting.
|The Red Cross says more than 40,000 people have fled their homes [AFP]
"Is your government's objective regime change in Georgia, the overthrow of the democratically elected government of Georgia?" Khalilzad asked Vitaly Churkin, the Russian ambassador.
Churkin said "regime change is an American expression. We do not use such an expression".
But he added: "But sometimes there are occasions, and we know from history, that there are different leaders who come to power, either democratically or semi-democratically, and they become an obstacle."
South Ossetia and Abkhazia split away from Georgia after fighting in the early 1990s and have run their own affairs without international recognition.
The two separatist provinces have close ties with Moscow, while Georgia has deeply angered Russia by wanting to join Nato.
Georgia, whose troops have been trained by American soldiers, began an offensive to regain control over South Ossetia overnight on Friday, launching heavy rocket and artillery fire and air strikes that pounded the regional capital Tskhinvali. Georgia says it was responding to attacks by separatists.
In response, Russia launched artillery shelling and air attacks on Georgian troops and entered South Ossetia to repel Georgia's attempt to retake the province.