Syrian President Bashar al-Assad vowed in an interview released today that he will "live and die" in his home country, as the crisis that has left at least 36,000 dead mounts to the point that the Red Cross says it is struggling to gain access to prisoners.

Assad spoke in English with Russia Today TV and excerpts of the interview were posted on the station's website on Thursday, with an Arabic voiceover.

Two days earlier, British Prime Minister David Cameron suggested that Assad could be allowed safe passage out of the country if that would guarantee an end to the nation's civil war.

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Syrian President Bashar al-Assad said he's not a "puppet" of the West, in an excerpt of an interview with Russia Today TV that was released Thursday. (Russia Today TV)

But Assad said in the interview: "I am Syrian, made in Syria, and I will live and die in Syria."

In excerpts of the interview, the Syrian president is seen casually talking and later walking with Russia Today's reporter outside a house, wearing a grey suit and tie. It was not clear where the interview took place.

"I am not a puppet, I was not made by the West for me to go to the West or any other country," Assad, 47, said.

The Syrian president also warned against foreign military intervention.

"I don't think the West is headed in this direction, but if it does, nobody can predict the consequences," he told the station.

The full interview will be broadcast on Friday, the TV station said.

Crisis 'deepening every day'

Earlier, Peter Maurer, head of the Geneva-based International Committee of the Red Cross, said it can't cope with the worsening civil war, despite an increase in the scope of the organization's operations since the crisis began in March 2011.

Maurer said Thursday "there is no doubt that the seriousness of the crisis is deepening every day. This trend has been uninterrupted since summer."

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A Syrian woman who fled from the violence in her own village carries a water bucket on her head in Atma, a Syrian village near the Turkish border, on Wednesday. (Khalil Hamra/Associated Press)

Since summer the Red Cross has improved its transportation and logistics, making it easier to bring in truckloads of food and medicine, but it has become overwhelmed by the dire need of hundreds of thousands of people struggling inside the country, he said.

He also said there has been no "major progress" on gaining better access to prisoners.

Maurer said the Red Cross is in "difficult negotiations" to expand the scope of their prison visits. As the arbiter of the rules of war known as the 1949 Geneva Conventions, the organization has a unique role to play in the monitoring of prison conditions worldwide and providing humanitarian help for people affected by conflict and armed violence.

The Red Cross is awaiting a response to request to visit 25 main prisons in Syria, but also is trying to gain access to other detention facilities, Maurer said.

One improvement, he said, was Assad's willingness to assign a senior Army officer within his military command to serve as a direct communications link with a Red Cross officer, giving the aid organization "a direct channel in terms of confidential conduct-of-hostility dialogue with the Syrian forces."

Mortar rounds land in Golan Heights

Meanwhile, Syrian strife continues to spill across its borders, with mortar rounds landing inside the Israeli-held Golan Heights, the third time in less than a week that the civil war has spilled over into the Golan.

Israel's military says three mortar rounds fired from Syria landed there on Thursday, but no injuries were reported.One round fell on a communal farm.

Security officials think the fire from Syria has been inadvertent.

On Monday, a bullet fired from Syria struck an Israeli military jeep on the Golan, a territory Israel captured from Syria in the 1967 Mideast war.

On Saturday, three Syrian tanks entered a demilitarized zone on the Golan.

Syria's daily death toll has been averaging 100 people or more recently, killed in clashes between rebels and troops, and in artillery shelling and regime airstrikes on rebel-held areas.

The Britain-based activist group Syrian Observatory for Human Rights, which relies on information from activists on the ground, said it has received reports of fresh fighting in the Damascus suburbs and in the neighbourhood of Souseh in the capital on Thursday.

It also said there were heavy clashes between anti-government gunmen and troops in northern Idlib province and in Aleppo, Syria's largest city, which has been a major front in the civil war since the summer.