The BAMS-D is a RQ-4A Global Hawk High-Altitude, Long, Endurance (HALE) drone that can carry out surveillance and reconnaissance missions over vast ocean and coastal regions, according to the US military.
Iran's Foreign Ministry issued a condemnation of the purported violation of its airspace and warned of a strong reaction to what it called a "provocative" act by the US.
White House press secretary Sarah Sanders said Trump was briefed Wednesday night and again Thursday morning about the incident.
Later, President Trump tweeted: "Iran made a very big mistake!" Trump's tweet came as his national security officials were huddling to weigh possible responses, CNN reported.
The incident comes at a time of escalating tension between the two countries.
On Monday, the US defence department said it was deploying 1,000 extra troops to the region in response to "hostile behaviour" by Iranian forces. It has already sent an aircraft carrier strike group and B-52 bombers.
The US has also accused Iran of attacking two oil tankers with mines last Thursday just outside the Strait of Hormuz, in the Gulf of Oman. Iran rejects the allegation.
It was the second time in a month tankers had been attacked close in the region, through which a fifth of the world's oil passes each day.
Tensions were further fuelled on Monday when Iran announced its stockpile of low-enriched uranium would next week exceed limits it agreed with world powers under a landmark nuclear deal in 2015.
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