At a recent summit in Damascus the leaders were filmed discussing the US president and using a biblical reference to joke about his personality.
During a visit to a Greek Orthodox church the pair discussed the biblical story of Saul’s conversion to Christianity on the road to Damascus.
God is said to have struck Saul, a Pharisee who persecuted Christians, blind, before the future apostle was cured and converted to the new faith.
In the video, posted to Twitter by a Russian journalist, Mr Assad is heard to say: “If Trump also travelled on this road, it would fix him.”
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The leaders are both seen to laugh at the quip.
Mr Putin then urges his host to extend an invitation to the US president, adding: “He’ll come.”
“I’m ready,” says Mr Assad, to which Mr Putin replies: “I’ll tell him”.
Moscow is the Assad regime’s strongest backer in the Syrian civil war and the campaign against Isis.