They want to set up a "people's council" of unspecified good and worthy people to spearhead political reform before new elections are held and have blocked key intersections in Bangkok for weeks to press their case.
Policeman and protester killed in Bangkok clashes
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The protests have been marked by occasional small bomb blasts and gunfire in which 21 people have been killed and more than 700 wounded since the crisis began in November.
The crisis pits the mainly middle-class and southern anti-government demonstrators, who are backed by the royalist establishment, against the largely rural supporters of Yingluck and Thaksin in the north and northeast.
Both sides have armed activists and some pro-government leaders have called for Thailand to be divided in two, along north-south political lines, prompting talk of a possible civil war.
"As of now, there is no clear sign that (civil war) will happen," national security chief Paradorn Pattanatabutr told Reuters.
"There are those who think differently and respect the law who can no longer tolerate this ... The government must do everything it can to avoid confrontation and to prevent each side setting up stages or rallies near each other.
"If they can do that, there should be no incidents."
The standoff also raises the question of whether the military will step in, as they have many times before, most recently in 2006 to remove Thaksin, although the army chief has ruled out intervention this time.
Thaksin's enemies say he is a corrupt, crony capitalist who manipulates the masses with populist handouts and is a threat to the monarchy, which he denies.