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.
"Jill and I join the families and friends of our fallen - and Americans across the country - in grieving the loss of these warriors in this despicable and wholly unjust attack," Mr Biden said in a statement.
The names of the servicemen killed and injured have not yet been released as officials work to notify their families.
On a visit to South Carolina, Mr Biden said "we had a tough day last night in the Middle East. We lost three brave souls".
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.
AzVision.az
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