Several southern Syrian towns have surrendered to the forces of President Bashar al-Assad, after more than a week of fierce attacks forced tens of thousands of people to flee their homes.

The United Nations says fighting has driven 160,000 people from their homes, towards borders with Jordan and the Israeli-occupied Golan Heights. Both countries say they will not allow any Syrians to cross.

Rebel fighters are trying to negotiate with Assad’s backer, Russia, as the opposition-held enclave in southern Daraa province rapidly shrinks under airstrikes and ground attacks, but say conditions offered are “hard to accept”.

Daraa is seen as the cradle of the Syrian uprising against Assad, which led to the seven-year civil war, so recapturing it would be both a strategic and propaganda victory for the government.

Fighting, which began on 19 June, continued into Saturday, a second week of conflict that the UN’s Syria envoy has said could match the siege of Aleppo and last year’s assault on eastern Ghouta combined.

The fighting in effect rips up a ceasefire agreed last year between Russia, the US and Jordan. It designated the area around Daraa as a “de-escalation zone” and Washington promised a strong response to any campaign that violated the agreement.

However, US forces have not taken action to stop attacks, and opposition spokesman Ibrahim Jabawi accused them of having struck a “malicious deal” to stay silent.

Syrians wait at the border areas near Jordan after they fled from the ongoing military operations Photograph: Anadolu Agency/Getty Images

Jabawi said rebel civilian and military officials held meetings with Russia to discuss surrender, but were offered terms they could not accept. Russia demanded the handover of weapons and said anyone who had borne arms would face trial.

Since Russia joined the civil war in 2015, it has helped turn the tide of the conflict with decisive but often brutal interventions, condemned by international agencies, aid groups and human rights monitors.

Government forces and their allies have repeatedly hit civilian targets including hospitals and schools, used chemical weapons and starvation sieges, in their battles to take over opposition-held areas around the country.

Rebel-held areas have been slowly recaptured, and the south-west is one of just three remaining strongholds still outside government control, along with Idlib in the north-west, alongside territory held by Kurdish forces in the north-east.

Even as the rebels negotiated with Russia for some limited protection from Syrian government troops, those forces were taking other towns in the area. Large parts of the rebel enclave has already fallen, and army advances near Daraa city could cut the opposition-held territory in two.

State television said at least five towns had agreed to surrender and accept government rule, and one had already raised the national flag, news agencies reported.

The Syrian Observatory for Human Rights says more than 100 civilians have been killed in airstrikes and shelling since the campaign around Daraa started. Tens of thousands of other civilians forced from their homes are gathered in extremely harsh conditions along the Jordanian and Israeli borders, some with tents but many sleeping in the open air.

The fighting has been so intense that aid convoys across the border have been halted, Jan Egeland, a humanitarian adviser to the UN said this week, begging Jordan to reopen its border to those fleeing conflict.

“I hesitate, in a way, to ask a small and poor county, who has more than a million Syrians on their soil, to take more,” he said. “But I have to do it, because these are women, children, civilians, who have no other place to flee if they are to escape the war zone.” Israel should also be asked to offer refuge, he said.