The White House and the US Defence Department did not immediately respond to Reuters requests for comment.
Several hours after the post, there was no evidence Kharg had been attacked.
Hamid Bovard, CEO of the National Iranian Oil Company, denied the key oil hub was being destroyed and operations had not stopped.
"Trump's tweets are laughable and conditions in Kharg are calm and appropriate," Nournews cited Bovard as saying.
"Kharg Island being blown to smithereens!!! President DJT," Trump said on social media, but gave no further details.
Attacking Kharg, which handled 90 per cent of Iran's oil exports before the war launched by US and Israeli attacks on February 28, would severely disrupt the Islamic republic's energy trade and put huge pressure on its economy.
Iran responded to the earlier US attack on rocket launchers on its island of Larak, about 660km east of Kharg, with an assault on two US bases in Jordan, Iranian media said.
The return to hostilities followed weeks in which the Trump administration has stepped up efforts to punish Tehran and its business partners with costly economic sanctions, shifting away from military options.
US Treasury Secretary Scott Bessent told Reuters on Sunday new secondary sanctions were likely to be unveiled weekly in the effort to pressure Iran, with an initial focus on banks.
"We're starting with the banks, and we're telling the banks it's not okay to have Iranian money and to aid the regime," Bessent said in an interview.
After imposing penalties on the United Arab Emirates branches of Egypt's Banque Misr on Friday over alleged financial links to Iran, Bessent said the next step may be cutting off an institution entirely from the dollar-based financial system.
"You're going to see a lot more of these every week," Bessent told Reuters, ahead of a meeting of Group of 20 finance leaders.
In describing the first known US military strikes since late July, a US official said forces attacked two Iranian launchers on Larak Island, more than 600km from Kharg.
The small island in the Strait of Hormuz near the strategic port of Bandar Abbas overlooks shipping lanes at the mouth of the Gulf, giving it a key role in tension over Iran, maritime security and global energy supplies.
"Islamic Revolutionary Guard Corps forces were observed preparing to launch rockets with sea mines into the Strait of Hormuz," the source added.
In retaliation, Iran launched ballistic missiles at Jordan, targeting two US bases. Jordan's armed forces had intercepted eight missiles that entered its airspace, state television said.
Iran's Revolutionary Guards said the attack killed and wounded several soldiers and citizens and vowed a "response and punishment" from Tehran, according to Iranian state media.
In a statement, the US Central Command said it took "limited, precise" action against mine laying forces of Iran's Revolutionary Guards that posed an "imminent threat" in the Strait of Hormuz. It gave no details.
After the US strikes, Iran attacked US forces station in Jordan, a Fox News reporter said on Sunday, citing a US source. Nearly all incoming missiles "have been intercepted at this point," the reporter said on X, flagging a lack of significant impact.
Amid threats to shipping, the number of visible commodity vessels transiting the Strait of Hormuz dropped to five a day over the weekend, shipping data showed.
The actual number could be higher as some vessels have switched off automatic identification systems to evade attacks.
Last week, Trump said all mines had been detonated or removed from the international waters of the strait and Iran had been warned that any vessel placing new mines would be destroyed.
