US President Joe Biden said the attack was carried out by “radical Iran-backed militant groups”. He added: “We shall respond.”
Iran has denied any involvement in the attack.
It is the first time that a strike has killed US troops in the region since Hamas’s 7 October attack on Israel.
Jordan says the attack took place in Syria, not inside Jordan.
There have been other attacks on US bases in the region, but before Sunday there were no fatalities, according to the US military.
It is not clear who is behind this latest attack.
President Biden said the US “will hold all those responsible to account at a time and in a manner of our choosing”.
Iran denied US and British accusations that it supported militant groups blamed for the strike.
“These claims are made with specific political goals to reverse the realities of the region,” foreign ministry spokesman Nasser Kanaani said, according to Tehran’s official IRNA news agency.
The White House said Mr Biden was briefed Sunday morning on the attack by US Defence Secretary Lloyd Austin and other officials.
US officials say at least 34 military personnel were being evaluated for possible traumatic brain injury, and that some of the injured soldiers were medically evacuated from the base for further treatment.
They also say that the drone struck the living quarters, which, if confirmed, could explain the high number of casualties.
US Central Command and President Biden said the attack was on a base in northeastern Jordan, near the Syrian border. It was later named by US officials as Tower 22.
A Jordanian government spokesman, Muhannad Moubaideen, however, told state TV that the attack targeted al-Tanf base in Syria.
In December, US officials said that US bases in Iraq and Syria had been attacked at least 97 times since 17 October.
Last month, the US carried out airstrikes against Iran-affiliated groups after three US servicemembers were injured, one critically, in a drone attack on a base in northern Iraq.
Earlier in January, one retaliatory US strike in Baghdad killed a militia leader accused of being behind attacks on US personnel.
In a pre-recorded interview with ABC News that aired on Sunday morning, Chairman of the Joint Chiefs of Staff Gen CQ Brown said that America’s aim in the region is to “not have the conflict broaden”.
“The goal is to deter them and we don’t want to go down a path of greater escalation that drives to a much broader conflict within the region.”
US and coalition troops are also stationed in the Red Sea after the Iran-backed Houthis began attacking commercial ships in the region. The Yemen-based group says it is targeting vessels in the region in support of Palestinians in Gaza, where Israel is fighting Hamas.
The US military has previously said “these unlawful actions have nothing to do with the conflict in Gaza”.
“The Houthis have fired indiscriminately into the Red Sea, targeting vessels impacting over 40 countries around the world,” Centcom has said.