The UN has urged both sides to "step back from the brink" and find a peaceful resolution.
Thai troops maintained a cordon around the protesters' main three-sq-km encampment at the Rachaprasong shopping district on Tuesday, where an estimated 5,000 red shirts, including hundreds of women and children, remained.
The authorities had warned the red shirts to leave by 3pm (08:00 GMT) on Monday, saying that those who remained faced two years in prison.
However, the protesters defied the order and the deadline passed without any action being taken.
Two negotiators spoke directly on the phone on Monday but neither side was willing to compromise on a demand that the other pull back first.
Nattawut Saikuwa, a protest leader, called Korbsak Sabhavasu, the government's chief negotiator, in the first direct contact between the two sides since Thursday's flare-up of violence.
Our correspondent said the five-minute talk was not very fruitful as both sides wanted the other to make the first move in pulling back.
But informal talks were continuing, according to the red shirts, although the government did not appear keen to confirm those talks, our correspondent said.
Red shirt leaders have proposed a ceasefire and talks moderated by the United Nations, a call the government has dismissed.
On Monday, the protesters said they would accept talks as long as a neutral arbiter took part and troops withdrew.
But Korbsak, who is a senior aide to the prime minister, Abhisit Vejjajiva, said "the government cannot entertain demands from the protesters".
"The best way forward is to stop talking about negotiation and for the protest leaders to call their people back to the Rachaprasong rally area and stop the violence," he said.
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The UN, meanwhile, called on both sides to negotiate an end to the violence.
Navi Pillay, the UN high commissioner for human rights, saying there was a high risk it could "spiral out of control" after the expiry of the Monday deadline.
"Ultimately, this situation can only be resolved by negotiation. I urge leaders to set aside pride and politics for the sake of the people of Thailand," she said in a statement.
"To prevent further loss of life, I appeal to the protesters to step back from the brink, and the security forces to exercise maximum restraint in line with the instructions given by the government."
The crisis appeared to be near a resolution last week when Abhisit offered to hold elections - a year early - in November, if the demonstrators left their protest area.
But hopes were dashed after red shirt leaders made more demands and Abhisit withdrew his offer.