Incoming shellfire was heard on the Afghan side by an AFP journalist around 9:30 a.m. local time (0500 GMT/UTC) after an interlude in fighting, the agency said.
The Torkham crossing has remained open for Afghans returning in large numbers from Pakistan, despite the land border being largely shut since fighting between the neighbors erupted in October.
Pakistani authorities said on Thursday that Afghan refugees who were waiting to return home from the Torkham border have been taken back to safer places following the latest clashes.
Pakistan in recent months has ramped up deportations, targeting over 700,000 Afghans who fled after the Taliban's return to power.
Both Afghanistan and Pakistan acknowledged fighting in border areas on Thursday, albeit disagreeing markedly on exact developments on the ground and the extent of each other's losses and gains.
Afghanistan's Defense Ministry said that operations were taking place along the border with Pakistan in five provinces.
The 2,611-kilometer-long border between the two countries is known as the Durand Line, which Afghanistan has not formally recognized.
The Taliban's deputy government spokesperson, Hamdullah Fitrat, claimed that "up to 55" Pakistani soldiers have been killed, with the remains of 23 of them brought back to Afghanistan, and that an undisclosed number have been captured.
The Taliban also claimed to have captured several Pakistani checkpoints in the border regions.
What Is Pakistan's Version of Events?
Pakistan has disputed Kabul's claims on casualties, captures and territorial losses.
The country's Information Minister Attaullah Tarar said that two Pakistani soldiers had been killed and three wounded.
"A total of 133 Afghan Taliban are confirmed killed, more than 200 wounded. Many more casualties estimated in strikes in Kabul, Paktia and Kandahar military targets." Mosharraf Ali Zaidi, a spokesperson for Pakistan's Prime Minister Shehbaz Sharif, said.
Pakistani Defense Minister Khawaja Asif said Pakistan's "patience has reached its limit" and said there is now "open war" with the Afghan Taliban.
The two sides had also reported cross-border fire on Tuesday, but without casualties.
Why Are Pakistan and Afghanistan Fighting?
Long-testy relations between the neighbors deteriorated sharply in recent months, hitting a low point with the deadly fighting in October that killed more than 70 people on both sides.
Land border crossings have been largely shut since.
Several rounds of talks followed an initial ceasefire brokered by Qatar and Turkey, but no lasting agreement has materialized in the conflict.
Saudi Arabia this month managed to negotiate the release of three Pakistani soldiers captured in October.
Islamabad accuses Afghanistan of failing to act against militant groups that it says also act out of Afghanistan, which the Taliban deny. Chief among these is the TTP, often referred to as the Pakistani Taliban, but Pakistan would also point to Baloch separatist groups from the border areas.
Pakistan's airstrikes followed a series of deadly suicide blasts, including an attack on a Shia mosque in Islamabad claimed by the Islamic State-Khorasan regional chapter that's primarily active in eastern Afghanistan.
Read: Afghan Taliban Attacks Met with Pakistani Strikes on Kabul
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