Opposition protesters have agreed to hold peace talks with the Thai government following days of deadly unrest in the capital.
But the government rejected the proposal, saying negotiations cannot start until the protesters disperse.
The proposal for peace talks also failed to stop sporadic fighting on the outskirts of Bangkok on Tuesday.
Earlier in the day, Al Jazeera's Aela Callan reported that gunshots and some explosions could still be heard in the city as clashes between opposition red-shirt protesters and government forces continued, but things were quieter than the day before.
A building was on fire and firefighters were having trouble accessing the site as it was in the live fire zone sealed off by the military, she said.
At least 37 people have been killed and 282 wounded since the latest confrontations started on Thursday, according to emergency services and local media.
The mediation proposal accepted by the red shirts but rejected by the government, 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, one of the red-shirt leaders, said.
But Satit Wonghnongtaey, a cabinet minister, quoted Abhisit Vejjajiva, the prime minister, as saying that "the situation will end only when the protest stops".
Separately, 40 other senators with more pro-government leanings called on the red shirts to surrender and face criminal charges, which in some cases include the death penalty for terrorism.
"It's just the beginning and it's the kind of an offer that doesn't carry much weight since the senators are not speaking in one voice," Somjai Phagaphasvivat, a political scientist at Bangkok's Thammasat University, said.
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.
Saikuwa called Korbsak Sabhavasu, the government's chief negotiator, in the first direct contact between the two sides since Thursday's flare-up of violence.
The UN has urged both sides to "step back from the brink" and find a peaceful resolution.
Navi Pillay, the UN high commissioner for human rights, said there was a high risk the violence could "spiral out of control".
Robert Amsterdam, Thaksin's lawyer, speaks
to Al Jazeera about the Thai crisis
"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."
Our correspondent 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," Callan said.
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.