Taliban fighters control 65 per cent of Afghan territory and are threatening to seize 11 provincial capitals, the EU believes.
A Brussels official said the Taliban were trying to cut off Kabul from forces in the north of the country that support the Afghan government.
The official said at a briefing in the Belgian capital that increasing numbers of Afghans were fleeing their homes, with many heading to Iran.
Delivering humanitarian aid to Afghans is becoming more difficult owing to the fighting between the Taliban and government forces.
Some countries have told their citizens to leave Afghanistan as the fighting escalates following the departure of US and Nato troops.
Military insiders said foreign embassies were hurrying to destroy spare weapons to stop them falling into the hands of the Taliban.
Several provincial capitals have already been overrun by the militants, who ruled Afghanistan until the US-led invasion removed them from power in 2001.
After withdrawing ground troops, the US launched air strikes to support government troops but said it was up to Afghan forces to defend their country.
Europe’s estimate that the Taliban have seized about two thirds of the country comes after the militants claimed to control 85 per cent of its provinces.
The EU official described the situation in Afghanistan as challenging but “not desperate”.
They said Brussels wanted to avoid Afghanistan from slipping into a state of civil war and triggering a “massive flow of migration” to Europe’s borders.
More than one million asylum seekers sought refuge in Europe during the 2015 migration crisis, many fleeing from Syria and Iraq.
Afghanistan’s situation was different to crises in Syria and Iraq because it has a solid government and recognised authorities, the official said.
But pressure is growing on EU governments to stop deporting people to Afghanistan because of the deteriorating situation.
UN human rights chief Michelle Bachelet said there were reports of acts that could amount to war crimes or crimes against humanity.
These included what she said were "deeply disturbing reports" of the summary execution of surrendering government troops.
"People rightly fear that a seizure of power by the Taliban will erase the human rights gains of the past two decades,” she said.
About 400,000 Afghans are estimated to have been displaced from their homes this year.
The country also faces a drought and a third wave of Covid-19 infections, said Antonio Vitorino, head of the International Organisation for Migration.
He said fighters and neighbouring countries should do everything possible to keep border crossings open to enable humanitarian assistance.
“These factors leave almost half of Afghanistan’s population in need of emergency relief assistance, with needs expected to continue to rise,” he said.
“I also call on the international community and our donors to support our efforts and not abandon Afghanistan at this critical moment.”