The PM is not budging, and on Sunday, protestors clashed with riot police.
Three people were also reported to have died in street clashes, as opposition supporters tried to break into key government buildings.
Protest leader Suthep Thaugsuban says the 'the people's coup' will be successful.
"We would like to demand that the government and the police think of the country, stop blaming and hurting the people, and return the power to the people so that people can use their sovereignty to solve the country's problems." he said.
Thailand specialist, Nicholas Farrelly, research fellow at the Australian National University says Khun Suthep is a veteran politician with a strong following in southern Thailand.
But even if Khun Suthep and his followers force the Prime Minister to resign, Dr Farrelly says their electoral prospects are still bleak.
Presenter: Sen Lam
Speaker: Dr Nicholas Farrelly, Research Fellow at the Australian National University
FARRELLY: Suthep is a provincial politician from the southern province of Surat Thani. He's been a controversial figure now in Thai politics going back very many decades. He was a minister in the democrat-led administration of Chuan Leekpai, back in the late 1990s, and then, of course, up until 2001, when deposed former Prime Minister, Thaksin Shinawatra came to power for the first time.
Suthep more recently, was in charge of national security during the prime ministership of Abhisit Vejjajiva, and during his long and distinguished and also somewhat fraught career, he's made a great number of enemies, he has a loyal, feverishly loyal support-base in some of the southernmost provinces, but he's a controversial figure and he's somebody whose prepared to get his hands dirty, in the way of Thai provincial politics, and here he is leading an insurrection.
LAM: He may be a controversial figure, but would it be accurate to describe him as an establishment figure as well?
FARRELLY: Suthep is definitely one of the Democrat Party's strongest campaigners. He's somebody who has managed now time in and time out to get himself in the mix, where the most important decisions are made.
He's somebody who has professed loyalty to the Royal Family and its influence in Thai society. He's somebody who's managed to keep a certain style of elite-driven politics right at the forefront. And so with that, he's somebody that various potent forces in Thai society look to, when push comes to shove and so here, they've got Suthep as the front man for this quite incredible showdown on the streets of Bangkok.
LAM: And it is indeed a showdown. The protesters seem to be upping the stakes. Where do you see all this heading? Do you think the military, for example, or the security forces are likely to view the protesters with some measure of leniency, given the background?
FARRELLY: It's always difficult to say in these kinds of contexts, exactly what the Thai armed forces have in mind. They've been relatively restrained up unto this point, although they have been mobilised in the last couple of days, to help reinforce some of the police lines around key government installations in Bangkok and elsewhere.
The armed forces appear to be looking to try to resolve this current crisis without reverting to what has been their habitual set of interventions, usually, of course, culminating in a coup.
The experience that they had, since trying to run the country in the 15 months after the coup of September, 2006, might give them some reasons to be more hesitant about the benefits that such an intervention would bring at this time. And so instead, they appear to be trying to broker some kind of resolution between the government of Prime Minister Yingluck on the one hand and the insurrectionist forces aligned with Suthep on the other.
Whether they can do that over the days ahead is really anybody's guess and my bottom line when it comes to these crisis threshold moments in Thai politics, is that you can never ultimately rule out the prospect of a coup.
LAM: The protestors, of course, are wanting to overthrow Prime Minister Yingluck Shinawatra, the sister of the former PM Thaksin. She's staying put, of course. But even if they did manage to persuade her to leave. If fresh polls were to be called, she might very well get back into office again,given that the Red Shirts have so much support outside the capital?
FARRELLY: So this is the quite incredible thing about the recent history of Thai politics. There were elections held in 2001, 2005, 2007 and then in 2011 and so on every occasion over more than a decade, the political parties allied to deposed Prime Minister Thaksin have come through very strongly. And I don't think there would be any reason to imagine that in an election held say in the early months of 2014, that we would see any different kind of result.
And that's why it's so difficult for Suthep and those who are out on the streets of Bangkok, calling for parliament to be dissolved and calling for the Yingluck government to step down. They don't have an electoral mechanism that they can confidently embrace in the hope of getting the reigns of power for themselves. Because if there is a poll, I think we'd all have to bet on a government or a former government spearheaded by former Prime Minister Yingluck Shinawatra, in such a scenario coming through and doing very well at the polls. And that just basically puts us back in the same situation yet again.
It's deja vu, but with this Thai style dynamic, which is dangerous, violent and in some cases at least, utterly unpredictable.