A photomontage, on his official website, appears to show the ex-US president playing golf in the shadow of a warplane or large drone.
The website image is captioned "vengeance is definite".
Twitter has suspended a small account which first tweeted the image.
A spokeswoman told Reuters the account - @khamenei_site - was fake and violated Twitter's platform manipulation and spam policy.
However, the post was also retweeted by Ayatollah Khamenei's much larger Farsi Twitter account with more than 300,000 followers. It has since disappeared from that account.
In the tweet, written in Farsi, the word "vengeance" is in red and the rest of the post says: "Soleimani's murderer and he who ordered it will have to pay".
On the Ayatollah Khomeini's official website, the image is posted prominently. The accompanying text quotes comments he made on 16 December, again vowing revenge "at any time".
Twitter had been urged to act after several users pointed out what they saw as the inconsistency of banning Mr Trump but not the Iranian leader.
Twitter closed Mr Trump's hugely influential account earlier this month after he published posts that were widely considered to have encouraged violence that overwhelmed the US Capitol.
"How come this atrocious psychopath can openly call for the assassination of a former US president, and not be kicked out of Twitter?" one user wrote in English.
"Trump's banned but this is perfectly ok. Is this a joke?" another user wrote.