"If (Pakistani) rulers hand him over to America then we will target these rulers. If Pakistani courts cannot punish Davis, then they should hand him over to us," a spokesman for the Al Qaeda-linked Tehrik-e-Taliban Pakistan told Reuters.
The U.S. consulate employee has been jailed for shooting two men last month. Davis claims he shot them in self-defense, and that his detention is illegal because he has diplomatic immunity.
Pakistani government officials have avoided taking a definitive stand on Davis' legal status in the face of popular anger over the shootout. Thousands have rallied against Davis, demanding he be hanged.
"We will give exemplary punishment to the killer Davis," the Taliban spokesman told Reuters.
Police say their investigation found Davis committed a "cold-blooded murder" and that that's the charge they'll pursue in court. It hasn't helped that the government of Punjab province, where any trial would be held, is run by a party that is a rival to the one running the federal government.
Sen. John Kerry, D-Mass., is expected to arrive in Pakistan Tuesday to discuss the case with senior Pakistani officials.
He is expected to issue a statement of regret over the incident, though American officials would not confirm that. The embassy described Kerry's visit as a way for the U.S. to remind Pakistan of the strategic importance of their relationship.
U.S. officials have threatened to withhold billions of dollars in aid to Pakistan to get Davis freed.
The Associated Press contributed to this report.