By Kareem Khadder and Judith Vonberg, CNN

(CNN) -- At least 270,000 Syrians -- more than the population of Orlando, Florida -- have fled their homes in southwest Syria since the government launched an offensive against rebels more than two weeks ago, according to the UN Refugee Agency (UNHCR).

Speaking to CNN on Monday, UNHCR spokesman Mohammad Hawari said that the number had "exceeded our expectations of 200,000."

The refugees are heading to the borders with Jordan and the Israeli-occupied Golan Heights, but neither country is willing to accept them.

The agency believes that the number of displaced people is likely to grow as fighting in the region continues.

Around 6.2 million Syrians have been displaced since the conflict began in 2011 -- and a further 6.3 million have fled the country as refugees, according to a recent UNHCR report.

The latest exodus began after the Russian-backed Syrian regime launched an offensive on June 19 to take Daraa province, located south of the capital Damascus. Syrian forces are wresting control from the Turkish-backed Free Syrian Army, as well as terror groups ISIS and Jabhat Fateh al-Sham.

At least 293 people, including rebel fighters, have been killed in the region since the offensive began, according to the Daraa Martyrs Documentation Office.

Seventy-five people, including 23 children and 11 women, were killed on Saturday alone as a result of clashes, Russian and Syrian airstrikes and artillery shelling, the group said Sunday.

Negotiations for an immediate ceasefire in southern Syria are ongoing, Jordan's Minister for Media Affairs and Communications, Jumana Ghunaimat, confirmed to CNN Monday.

Jordan and Israel are providing aid to the displaced but are refusing to allow Syrian refugees inside their countries. Jordan is already hosting 1.3 million Syrian refugees but closed its borders in 2016 after an ISIS suicide attack killed six of its border guards.

Speaking Friday, Jordanian Foreign Minister Ayman Safadi said that the resource-poor kingdom had "reached its capacity."

READ MORE: Syrian war: Tens of thousands flee toward closed borders

Amnesty International urged the Jordanian government to open its borders to those fleeing Syria, beginning with refugees who are sick or injured.

"Daraa residents are effectively trapped -- many of those who are displaced are living in makeshift tents in the searing heat with insufficient food, water or medical care, and with the constant fear of being exposed to attacks at any given point. Jordan's border is their only gateway to safety," said Lynn Maalouf, Amnesty International's Middle East Research Director.

Israel does not typically take in Syrian refugees, except to give them medical treatment. The country has delivered four shipments of aid to displaced people living in tent camps near the Golan Heights border, the Israel Defense Forces (IDF) said last week.

The deliveries included 300 tents, 13 tons of food, 15 tons of baby food, three pallets of medical equipment and medicine and 30 tons of clothing and footwear, according to the IDF.

The army reiterated in its statement Friday that the country "cannot permit Syrians fleeing hostilities to enter Israel, and will continue maintaining Israel's security interests."