Jatuporn Prompan, a key red-shirt protest leader, said "as long as the troops are confronting the people, the people will fight."
A mediation proposal, accepted by the protesters but subsequently rejected by Abhisit, was floated by a group of 64 senators in the 150-member upper house.
"We have agreed to take a new round of talks proposed by the senate because if we allow things to go on like this, we don't know how many more lives will be lost," Nattawut Saikua, a red-shirt leader, said after 38 people were killed and nearly 300 others were wounded in the fighting.
The authorities had warned the red shirts to leave their protest site 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.
Navi Pillay, the UN high commissioner for human rights, called on both sides to negotiate an end to the violence, saying there was a high risk it could "spiral out of control" after the expiry of the Monday deadline.
"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," she said in a statement.
Robert Amsterdam, Thaksin's lawyer, speaks
to Al Jazeera about the Thai crisis
The violence turned parts of the city of 12 million into no-go zones as troops used live ammunition against protesters, who fought back mainly with homemade weapons.
The red-shirts consider Abhisit's government illegitimate because it came to power with the backing of the army, in a 2008 parliamentary vote after a court ruling ousted elected allies of Thaksin Shinawatra, the deposed Thai prime minister.
Al Jazeera's Aela Callan said international mediation will not be easily accepted.
"Thais are notoriously reluctant to accept international mediation, in fact on the whole they resent 'meddling' as they call it, in their affairs," she said
The crisis, which began when rolling demonstrations were launched in mid-March, has now left 67 people dead and about 1,700 wounded.
Twenty-five people died in a failed army crackdown on April 10.