"The UN is trying today, for the third day running, to get wounded civilians, including 50 children, out of that pocket.
"But it depends on clearance from the Sri Lankan government, a lull in the fighting, and permission from the Tamil Tigers. The crisis is growing day by day."
Sri Lankan government forces have the Tigers in a stranglehold, Birtley said on Thursday.
"The army and air force will use its aerial power and reconnaissance ability when they move into thick jungle around Mullaittivu," he said.
"At the moment, the government forces are fighting on conventional lines, as we understand. We have seen a number of army casualties. The army maintains it is inflicting more damage on the LTTE but we only have their word for it."
Thursday's fighting comes a day after a senior district health official said that up to 300 innocent people have died and more than 1,100 have been injured in the crossfire.
"It is high time to take decisive action and stop further bloodshed because time is running out"
Jacques de Maio, the head of Red Cross operations for South Asia
"There are a lot of deaths - more than 250 to 300," Thurairajah Varatharajah, the senior government health official in the Mullaittivu district, told the AP news agency on Wednesday.
He said at least 248 of the injured civilians in the district's three hospitals were younger than 15 years old.
"There are dozens of unclaimed bodies lying in the hospital mortuaries because no relatives are coming forward to claim them," he said.
The Sri Lankan government has insisted that there have been "zero civilian casualties" in its operations.
Jacques de Maio, the head of Red Cross operations for South Asia, said on Wednesday: "People are being caught in the crossfire, hospitals and ambulances have been hit by shelling and several aid workers have been injured while evacuating the wounded.
"It's high time to take decisive action and stop further bloodshed because time is running out."
The Red Cross has appealed to both sides to allow civilians to leave the combat zone.
Sri Lanka's military said the LTTE, which are fighting for the creation of an independent Tamil state in northern Sri Lanka, wanted to create a "last-minute civilian tragedy" because the army was about to completely defeat them.
Udaya Nanayakkara, a military spokesman, said claims of civilian deaths were part of a "cheap propaganda exercise" by the LTTE.
Pranab Mukherjee, India's foreign minister, on a surprise visit to Sri Lanka on Tuesday, called for urgent measures to protect civilians caught up in the island's ongoing fighting.
Sri Lanka's minority ethnic Tamils have close cultural and religious links with the 55 million Tamil population in the south Indian state of Tamil Nadu.
Sri Lankan forces pushed into the northern village of Visuamadu, held by the LTTE, on Wednesday, a defence ministry official said.
The ministry claimed that Tamil separatists retreated their long-range artillery guns into an area declared a 35 square km "safe zone" for civilians, and were firing at the military from there.
The government has said civilians are being forced to move with the Tigers to act as human shields, but there has been no independent confirmation that is the case.
The Red Cross' De Maio said: "When the dust settles, we may see countless victims and a terrible humanitarian situation unless civilians are protected and international humanitarian law is respected in all circumstances."
The United Nations has issued similar warnings, with Ban Ki-moon, the UN secretary general, saying he was "deeply concerned" for the safety of civilians caught in the fighting.
The official death toll is unknown as medical officials have said they have been ordered by the health ministry in Colombo not to reveal casualty figures.
A fresh offensive against the LTTE was launched by the government late last year, shrinking the northern territory under its control.
The military seized Mullaittivu on Sunday, which they claim was the LTTE's last urban centre.
Tens of thousands of people have been killed in Sri Lanka since the LTTE launched its war in 1972.