Reporting from Aleppo, Syrian state TV correspondent Shadi Halawi said the city's international airport was "100 percent safe," although he acknowledged that rebels were able to enter the "Brigade 80" base. Government troops were pushing them out, he said.
Rebels control many areas in the north and east of the country, and hold whole neighborhoods in Aleppo, Syria's largest urban center and its main commercial hub. The government maintains a tight grip on Damascus, and several central provinces, including Homs and Hama.
For nearly a week, rebels have been trying to slowly battle their way into the capital Damascus from neighborhoods and towns on its doorstep, and have punched to within a mile of the heart of the city.
There was more fighting in the capital on Tuesday. Activists said government warplanes struck opposition strongholds in several suburbs including Zamalka and Douma.
In a rare attack, a mortar round fell outside the Homs oil refinery, killing an employee and wounding three. In addition to Homs, Syria has another oil refinery in the coastal city of Banias.
Even as the fighting intensifies, the opposition and the government are saying they are ready for a dialogue that they hope would end the crisis.
Syrian Information Minister Omran al-Zoubi was quoted by Syrian TV as saying that "the fate and future of Syrians should be decided in Damascus and by the Syrians themselves."
He was apparently responding to calls by opposition leader Mouaz al-Khatib, who said he is ready for talks with the regime in Syria as long as it is held in rebel-held territories. Al-Khatib also demanded that authorities release 160,000 detainees and renew passports of dissidents living abroad.
National Reconciliation Minister Ali Haider told reporters in Damascus that the government has fulfilled a package of measures to renew the passports of Syrians abroad and they have been put in force.
Also Tuesday, Assad attended the first meeting of the Syrian Cabinet after a minor government reshuffle two days ago. He said in comments carried on the presidency's Facebook page that the world now recognizes how steadfast his regime really is.
In Turkey, Interior Minister Muammer Guler said the death toll from an attack Monday on a border crossing between Syria and Turkey had risen to 14 from 13.
The frontier area has seen heavy fighting, although attacks on the crossings that are used by Syrian refugees and international aid agencies have been rare.
A major Syrian opposition faction accused the government of the bombing, saying it narrowly missed 13 leaders of the group, including its president.
Syrian National Council President George Sabra told reporters in Turkey on Tuesday that the car bomb went off shortly before the SNC president and 12 other members of the group's executive council drove through a border crossing with Turkey.
"We heard an explosion nearly half an hour before we reached the border crossing point," Sabra said.
There was no claim of responsibility for the blast. Turkey's deputy prime minister, Besir Atalay, said Tuesday that a preliminary investigation indicated three assailants parked a car packed with explosives in no man's land between two border gates, then detonated it remotely about 20 minutes later.
Associated Press writer Suzan Fraser in Ankara, Turkey contributed to this report.