US President Donald Trump has approved the execution by firing squad of Nidal Malik Hasan, the former US Army psychiatrist convicted of killing 13 people at Fort Hood, Texas, in 2009.

The Pentagon announced on Monday that Defence Secretary Pete Hegseth had recommended the firing squad and that Trump approved the decision.

Pentagon spokesman Sean Parnell said the Secretary of the Army would determine the date and location of the execution.

Hasan was sentenced to death by a military jury in 2013 after being convicted of carrying out the November 5, 2009, attack, which also left more than two dozen people wounded.

If carried out, the execution would be the first by the US military since 1961.

Hasan opened fire at a medical facility on the sprawling Fort Hood military base, which was a major staging point for US troops deploying to Iraq and Afghanistan.

Twelve of those killed and 30 of those wounded were soldiers. Hasan was shot by a civilian police officer who responded to the attack and was left partially paralysed.

During his court-martial, prosecutors said Hasan opposed the US military's operations in Iraq and Afghanistan and had communicated online with Anwar al-Awlaki, an American-born radical cleric linked to al-Qaeda who was later killed in a US drone strike in Yemen.

Hasan, a US-born Muslim of Palestinian descent, represented himself during the trial. He called no witnesses, gave no testimony and did not contest the charges. His court-appointed lawyers had raised concerns that he was seeking execution.