One was set off near the bell tower of St Mark’s Basilica, the ANSA news agency reported, while the other was in the centre of the square.
A picture of the scene showed smoke covering much of the square.
The Gazzettino newspaper said it was a tourist who alerted police.
He had been sitting outside a cafe when he saw one person set off a smoke bomb, while another two tried to enter the store.
— Nicole DeMentri (@nicoledementri) March 17, 2017
Carabinieri and State police were said to be quickly on the scene, with the robbers fleeing before they could carry out their heist.
Nicole DeMentri filmed the smoke rising into the air.
“The Bell Tower in San Marco Square is filled with smoke as visitors rush out of the doors,” she wrote on Twitter. “The smoke appears to be going upwards.”
Shortly afterwards, she wrote: “Minutes later the smoke seems to have stopped, but visitors seem not to be allowed inside and instead crowd in the square.”
Witnesses said they feared they may have been a terrorist attack.