ISLAMABAD—Pakistan on Monday denied that a key Afghan insurgent group was using it as a base, responding to statements by the U.S. national security adviser about the group’s links to Pakistan during a weekend visit to Islamabad.

Washington and Kabul believe the group, called the Haqqani network, is based in Pakistan and is behind a series of bombings in the Afghan capital in August. U.S. security officials also believe the group is responsible for many of the most deadly and complex attacks in Afghanistan over the past decade.

...