The bounty offer was posted by Thaksin's son Panthongtae on his Facebook page.
A video clip showing a man hurling a grenade at demonstrators at Victory Monument was shown on Monday at a Centre for the Administration of Peace and Order (Capo) press conference.
"Ten million baht cash is a bounty offered by my father on the attacker. This is in addition to the previous amount of 10 million baht offered for information on those who set fire to the Central World," Mr Panthongtae said.
Mr Panthongtae said the face of the man in the clip is clearly visible, as was a whistle dangling from his neck.
He argued that someone will identify the attacker in due course, since the image in the video clip is so clear, but warned information on the suspect might not come quickly enough because some want more people to be killed and wounded, to prompt military intervention in the current political conflict.
"I asked my father to offer an additional amount of money so that the truth could emerge soon," he said.
Mr Panthongtae added that he believes it is clear that anti-government protest leaders were the behind the blasts that wounded and killed their own supporters.