The bill has also drawn opposition from some Thaksin supporters who oppose immunity for Democrat Party leader and then-Prime Minister Abhisit Vejjajiva and his then-deputy for their alleged roles in the deaths of scores of protesters in a bloody 2010 crackdown.
The original draft of the bill, approved in principle by the lower house in August, did not extend the amnesty to the leaders of the pro- and anti-Thaksin groups, but a committee in mid-October changed the bill to include them.
Since it was passed by the lower house last Friday, the bill has set off daily demonstrations that have drawn tens of thousands of protesters onto Bangkok's streets and put heavy pressure on the government and the Senate.
More than 1,000 protesters on Thursday gathered at a busy intersection in downtown Bangkok to blow whistles in protest of the bill, while thousands of others marched from a university campus to submit an anti-amnesty petition to the Senate.
The Cabinet previously enacted a special security law in three Bangkok districts to bar demonstrators from entering areas surrounding the prime minister's office and Parliament.
Yingluck Shinawatra assured protesters in a televised address Thursday that other similar versions of the bill are not in the pipeline if the Senate strikes down the contentious amnesty bill.
"Now that every side has announced their will, including the House of Representatives that withdrew other drafts of the bill, I'm begging you, brothers and sisters, to call off the protests," Yingluck said.
She said the bill was aimed at helping ordinary people and those affected by the 2006 coup, not at getting rid of corruption cases.
Thailand's anti-graft commission said this week that the amnesty bill could impact more than 25,000 corruption cases it is investigating.