Zalmay Khalilzad's comments come as renewed fighting for days has plagued Afghanistan's southern Helmand province.
The Taliban pledge came after a meeting with U.S. peace envoy Zalmay Khalilzad and U.S. Gen. Austin Miller.
The announcement of U.S. strikes comes after a gun battle on Monday was reported in and around Lashkar Gah.
U.S. President Donald Trump has vowed to bring American troops home by Christmas.
Taliban spokesman Zabihullah Mujahed said Trump's statement was welcome and he considered it a positive step.
The Taliban are pushing for a return to power, having signed a landmark troop withdrawal deal with Washington.
The Taliban have continued attacks even as peace talks with the Kabul government are underway.
The airstrikes come as Taliban and government-appointed negotiators are meeting for the first time in Qatar.
The embassy didn't provide specifics, including how imminent the threat is.
The U.N. Security Council unanimously adopted a resolution welcoming the start of negotiations.
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