The Syrian Observatory for Human Rights said at least 20 people were wounded in the explosion near Souq al-Hamadiyeh district, and that six of the dead were Lebanese citizens.
The Lebanese agency that organised their trip gave the same death toll.
Syrian state media, which reported a toll of six dead and 19 wounded, said the blast was caused by an explosive device rather than by a suicide bomber.
Officials had found and defused a second bomb that had been placed inside the bus before it detonated, said the official SANA news agency.
Al-Nusra Front, the affiliate of Al-Qaeda in war-ravaged Syria, claimed responsibility for the attack in a statement posted online.
Observatory director Rami Abdel Rahman said the bus was carrying Shiite Muslim pilgrims on a tour of religious sites.
In Beirut, an official with the Lebanese group that organised the trip said the passengers on the bus were Lebanese.
"They set out from Beirut at 5:30 am (0330 GMT) this morning," Fadi Khaireddin told AFP, adding that the bus had space for 52 pilgrims, as well as the driver and trip administrator.
"The bus is usually full," he added, though he could not confirm how many people were on the trip this weekend.
The bus had made its first stop at the Sayyida Roqaya shrine and was heading towards the revered Sayyida Zeinab shrine in southeast Damascus when the attack occurred.
- Hezbollah slams 'barbarity' -
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The agency had been making regular trips to Syria despite the civil war there, with groups leaving each weekend for a day-long visit to shrines revered by Shiites across the border.
Syrian state television showed footage from the scene of the blast, with men in military uniforms picking through the wreckage of the bus.
Its front half was mostly blown off, leaving only the metal frame, and bags of belongings were strewn across the remaining seats.
The channel also showed images from inside a hospital where the wounded were treated, including a woman whose black robes had been lifted up, revealing a bloodsoaked undershirt.
The Lebanese Shiite movement Hezbollah denounced the attack.
This "is part of the series of explosions that targets pilgrims in Syria, civilians in Iraq, believers in Pakistan" and "proves the barbarity of the terrorists", it said in a statement.
Parts of Damascus have remained relatively unscathed by the fighting raging across much of Syria since an uprising erupted in March 2011.
But rebels regularly fire rockets into the capital from rear bases in the surrounding countryside, and the city has also been hit by bombings.
Despite the conflict, the road from the Lebanese border to Damascus remains relatively safe, and Lebanese Shiite pilgrims have continued to visit religious sites in Syria.
More than 200,000 people have been killed in Syria since the conflict started, and around half of the country's population has been displaced.