One of the 28 wounded is Sithinee Huangnak, a reporter for Post Today newspaper, who was admitted to nearby Ratchawithi Hospital. Ms Sithinee was later transferred to Phyathai Hospital 2 hospital.
The other wounded were rushed to Ramathibodi and Phra Mongkut hospitals, the reports said.
Bangkok Post quoted Thaworn Senneam, a former Democrat MP and core PDRC co-leader in charge at the Victory Monument rally stage, said an unidentified man threw an explosive device near a press centre tent behind the rally stage and ran off. He was chased by security guards and protesters, prompting him to toss another bomb.
But the group continued to try to catch him and the man fired a shot at them, leaving at least one person injured, before fleeing on a motorcycle driven by another man, heading toward Tuek Chai intersection.
It was believed the bomb thrower was one of six assailants at the scene. Mr Thaworn believes the bomber was targeting him as the bomb exploded only 20 metres away from where he was standing. Before the incident, guards spotted a suspicious-looking man and asked him to leave the protest venue.
A group of suspicious men later were seen scouting at the back of the rally stage, he said.
AFP quoted protesters as saying the injuries were caused by two separate blasts minutes apart.
"People were speaking on stage when there was a loud explosion," said Theerayuth Utakaintanond, 52.
"Almost two minutes later, I think, I heard another explosion and everyone started to run in panic."
There were blood-splattered clothes and a small crater at the scene of one of the blasts, according to an AFP reporter.
It is not yet known whether the explosive devices were grenades like the one used in Friday's attack.