'Conundrum'
Protest leader Suthep Thaugsuban, a silver-haired former deputy prime minister from the pro-establishment Democrat Party, has called for a "people's coup" to throw out the "Thaksin regime". He has proposed a vaguely defined "people's council" of appointed "good people" to replace the government.
Yingluck has rejected that as unconstitutional.
One analyst said the king appeared reluctant to get drawn into the mess.
"I believe the king understands the conundrum," said Thak Chaloemtiarana, a Thai academic at Cornell University in the United States. "He must also worry about his own influence and how far it can go."
The crown prince expressed his concern about the political unrest last week and urged people to settle differences peacefully. But that did not stop the violence.
"If the king did this and nothing happened, it could diminish his aura and legacy," said Thak.
In theory, Thailand's monarchy is above political division. The king is a constitutional monarch, with no formal political powers but immense influence.
But despite the king's silence over recent years of turmoil, the palace has been drawn in. Thaksin's supporters believe some of the king's advisers instigated the 2006 coup in which Thaksin was ousted.
They also cite Queen Sirikit's attendance at the funeral of an anti-Thaksin protester killed in clashes with police in 2008. That fuelled suspicion of royal backing for Thaksin's opponents.
The divide between the poor and what they see as the establishment elite represents a collapse of a traditional order in Thailand at a time when people have begun to broach the hitherto taboo topic of succession.
The crown prince does not command the same devotion his father does and some in the establishment elite are nervous about the prince's suspected links with Thaksin in the past and the influence that that connection could give Thaksin.
The crown prince's recent appeal for people to settle differences peacefully was issued through Bangkok police chief Kamronwit Thoopkrachang, a staunch Thaksin ally.
"But getting rid of this government is not the answer," said Thak. "The genie is out of the box and the old power elite must change and learn how to live with current political realities."