The Pakistani government has given 1.7 million Afghans it says are living illegally in the country until November 1 to leave voluntarily or be forcibly removed.
More than 130,000 people have left Pakistan since the order was given at the start of October, according to Pakistani border officials, creating bottlenecks on either side of crossing points.
Taliban authorities thanked Pakistan and other countries that have hosted millions of Afghans who fled during decades of conflict.
However, in a statement late Tuesday, they also "asked them to not forcibly deport Afghans with little notice but to give them time to prepare".
Since taking power, the Taliban government has urged Afghans to return home but has also condemned Pakistan's actions, saying nationals are being punished for tensions between Islamabad and Kabul.
Pakistan has said the deportations are to protect its "welfare and security" after a sharp rise in attacks, which the government blames on militants operating from Afghanistan.
The Taliban government statement again denied the claim, saying: "In countries where Afghans live, they have not threatened the security of those countries, nor have they been the cause of instability."
The statement hit out at Pakistan for restrictions on what Afghans could bring across the border, including property such as livestock and cash.
Border officials on the Afghan side at the Torkham crossing in eastern Afghanistan said they were facing an "emergency situation" as they tried to keep up with waves of arrivals in their thousands.
An ad hoc settlement has sprung up near the border post, where people are becoming increasingly desperate, sleeping outdoors with limited access to food, water and medicines as they wait for registration.
The government has established a High Commission to address the issue and said two temporary camps would be set up in the area near Torkham.
Wednesday's statement also urged wealthy Afghans to work with the High Commission to support returnees with transport, accommodation and shelter.
Officials have also said staff, technical reinforcements and trucks carrying mobile toilets, generators and water tankers were being deployed to Torkham.
A high-level government delegation visited Torkham on Tuesday, pledging support to returnees who had been "forcibly evicted by the Pakistani government against all the norms, good neighbourliness and humanitarian sentiments".