No images have been released of Khamenei since an Israeli strike at the start of the war that killed much of his family, including his father and wife.
His first comments came in a statement read out by a television presenter on Thursday. In the statement, he vowed to keep the Strait of Hormuz shut and called on neighbouring countries to close US bases on their territory or risk Iran targeting them.
"We know the new so-called not so supreme leader is wounded and likely disfigured. He put out a statement yesterday. A weak one, actually, but there was no voice and there was no video. It was a written statement," Hegseth told a briefing on Friday.
"Iran has plenty of cameras and plenty of voice recorders. Why a written statement? I think you know why. His father - dead. He's scared, he's injured, he's on the run and he lacks legitimacy."
An Iranian official told Reuters on Wednesday the newly appointed supreme leader was lightly injured but was continuing to operate, after state television described him as war-wounded.
During the briefing, Hegseth said the United States would show no mercy in the war.
"We will keep pressing, keep pushing, keep advancing. No quarter, no mercy for our enemy," he said.
"No quarter" is the refusal to spare the life of someone who has expressed their intention to surrender - something prohibited by law.
"International humanitarian law prohibits the use of this procedure, that is, ordering that there shall be no survivors, threatening the adversary therewith, or conducting hostilities on this basis," according to the International Committee of the Red Cross (ICRC).
The US has carried out strikes against more than 6000 targets in Iran over the past 14 days.
Two US officials, speaking on the condition of anonymity, said the Pentagon was sending additional forces to the region, including the USS Tripoli along with its Marine expeditionary unit. The amphibious assault ship, currently in the Asia-Pacific region, is capable of carrying fighter jets as well.
The Pentagon has previously said additional troops would be heading to the region.
Hegseth's claim came after President Donald Trump said the US is going to be hitting Iran "very hard over the next week", shortly after issuing a partial 30-day waiver for purchases of sanctioned Russian oil, hoping to ease prices fuelled by the US-Israeli war on Iran.
Prices have been whipsawing on Trump's changing comments on the likely duration of the war, which has prompted Iran to attack vessels in the Strait of Hormuz, the conduit for one-ifth of the world's oil.
Trump has previously said the war is "complete", and also promised to guarantee the safety of vessels in the strait. In a Fox News interview aired on Friday, Trump said the US would escort shipping there "if we needed to".
Benchmark Brent crude eased about 0.6 per cent to around $US99.80 ($A142.32), still up almost 40 per cent since the start of the conflict.
After nearly two weeks of war, 2000 people have been killed, most of them in Iran, but many also in Lebanon and a growing number in the Gulf, which has for the first time in decades of Middle East conflicts found itself on the front line.
Meanwhile, the US is offering a reward of up to $US10 million ($A14 million) for information about senior Iranian military and intelligence officials, including its new leader Khamenei.
The reward targets 10 officials associated with Iran's Islamic Revolutionary Guard Corps (IRGC), according to the State Department website.
The military force, created after Iran's 1979 Islamic Revolution, is loyal to the supreme leader and tasked with protecting the Shi'ite clerical establishment.
