However Thida Thavornseth, a leader of the "red shirts", who support Mr Thaksin, told AFP news agency that Mr Suthep "wants to throw out democracy and replace it with an ultra-royalist administration".
The opposition Democrat Party has also started a censure motion in parliament against the government, over its alleged misuse of the budget.
The motion highlights an expensive rice subsidy scheme launched by the government after it took office.
Under the scheme, the government bought rice directly from farmers, paying more than the market rate. India and Vietnam increased their share of global rice exports as a result, overtaking Thailand as the world's largest rice exporter.
The government is expected to defeat the censure motion, since the ruling Pheu Thai party has a majority in parliament.
But, with a timid and poorly-trained police force, it is not clear how the government can reassert its authority in a city where a sizeable part of the population say they have lost all faith in their democracy, says the BBC's Jonathan Head in Bangkok.
The protests are the biggest to hit Thailand since the violence of 2010, when "red-shirt" opponents of the then Democratic Party government occupied key parts of the capital.
More than 90 people, mostly civilian protesters, died over the course of the two-month sit-in.
A government led by Ms Yingluck and the ruling Pheu Thai Party was subsequently elected and since then Thailand has remained relatively politically stable.
But the opposition accuse Mr Thaksin of running the government from self-imposed exile overseas, and the now-shelved amnesty bill has served as a spark for renewed protests.
Speaking in parliament on Tuesday, she said: "There are some accusations that I lack independence, and that I lack intelligence, and have to be controlled by pushing a button.
"I have to say, have I not been independent in the past two years that I administered the country as the head of the government?