"Afghanistan's borders with Tajikistan, Uzbekistan, Turkmenistan, and Iran, or about 90 percent of the border, are under our control," Zabihullah Mujahid told the state-run RIA Novosti news agency, a claim that could not be independently verified.
The militants are pushing across Afghanistan, snapping up territory, seizing border crossings and encircling cities, with the withdrawal of US and NATO troops all but complete.
The resurgent militants now control about half of Afghanistan's roughly 400 districts.
Mujahid told RIA Novosti the Taliban would not tolerate the Daesh [ISIS] militant group in Afghanistan.
"We assure you that we will not allow Daesh [ISIS] to become active in the country, in areas under our control," he said.
"There are no militants from Central Asia or China in the county," he added.
Ex-Soviet Tajikistan, on Afghanistan's border, held a large-scale military inspection on Thursday -- the first of its kind in the country's 30-year history.
The Taliban's offensives in recent weeks have forced Afghan refugees and government troops to make their way across the Tajik border.
Russia, which maintains military bases in Central Asia, said it would stage military drills with Tajikistan and Uzbekistan near the border with Afghanistan next month.