On Tuesday, Trump said Saudi officials had engaged in the "worst coverup ever" and that those behind the killing "should be in big trouble".
He told the Wall Street Journal: "Well, the prince [Mohammed bin Salman] is running things over there more so at this stage".
Erdogan refused to accept Saudi Arabia's explanation that the Saudi journalist was accidentally killed.
The well was at centre of speculation yesterday that it may contain body parts belonging to the journalist.
"All evidence gathered shows that Jamal Khashoggi was the victim of a savage murder". Erdogan also demanded that the Saudis tell him the names of every person involved.
This contradicts previous Saudi media claims that Khashoggi's death came by accident during a fistfight that broke out in the country's consulate in Turkey.
The phone call between Recep Tayyip Erdogan and Crown Prince Mohammed bin Salman was made at the latter's request, said the source, who spoke on condition of anonymity due to restrictions on talking to the media.
Despite the US Treasury Secretary Steve Mnuchin's withdrawal along with a number of high profile executives from the likes of Google, Uber, Mastercard and JP Morgan, Al-Falih said ties had not been damaged.
The secretary of state also told reporters at the State Department that the USA has identified some of the people "who we suspect to be responsible" for Khashoggi's death.
The death of Khashoggi, a US resident and Washington Post columnist, has sparked global outrage and threatened relations between Riyadh and Washington as well as other Western nations.
It comes on the day that Turkey's president openly accused Saudi officials of plotting Khashoggi's murder for days.
Saudi Arabia's prosecutor has determined the murder of Jamal Khashoggi at the Saudi consulate in Istanbul premeditated, contradicting earlier contentions that the journalist was killed by accident.
Trump repeated the denials by Saudi Crown Prince Mohammed bin Salman that he knew of the plot before it was carried out.
Mr Trump's criticisms of Saudi officials are his strongest so far, but he has continued to highlight the kingdom's importance as a USA ally.
Saudi Arabia's King Salman was also present, with photographers and a film crew, as the royals attempted to mend their image with publicised condolences to the grieving family.