In Saudi Arabia, the crown prince addressed the crisis publicly for the first time at an investors conference Wednesday, calling the killing a "heinous" crime that was a "very, very painful incident for all Saudis".
Meawhile, Rouhani urged the Turkish government to continue "impartial and precise investigations to unearth the fact for the world opinion".
Khashoggi's killing was the first issue the crown prince raised after he strode on to the stage at the Future Investment Initiative conference in Riyadh.
"The ideal thing for Trump to do is recognize that Mohammed bin Salman is weakened by the outrageous murder of Khashoggi", said Indyk, who argues that the one threat to the crown prince's survival is if the United States tells Saudi Arabia's King it can not work with his heir apparent.
Saudi Arabia's Foreign Ministry wrote on Twitter that the leaders shared "their deepest condolences and sympathy to the family of Jamal Khashoggi, may God rest his soul".
Saudi Arabia had insisted for weeks that the journalist walked out of the consulate, before changing their account to say he died in a brawl.
Turkish President Tayyip Erdogan spoke to Prince Mohammed on Wednesday and the two discussed the steps needed to bring to light all aspects of Khashoggi's death, a presidential source said.
He said the killing was planned in advance and called on Saudi Arabia to reveal more details, including where Khashoggi's body was.
For their part, the Trump administration and the US defence industry are scrambling to save the few actual deals in a much-touted $110 billion arms package for Saudi Arabia.
Trump said he wanted to believe Prince Mohammed when he said that lower-level officials were to blame for the October 2 killing at the Saudi mission.
"There are now those who are trying to take advantage of the painful situation to create a rift between the kingdom and Turkey", he said.
Hariri, who has Saudi citizenship, was summoned to Riyadh a year ago, briefly detained and compelled to resign as prime minister in a speech read on television until France intervened to free him.
While many in the Western political establishment have expressed shock that Saudi Arabia would target one of its journalists, the prospect of imposing economic sanctions or cancelling multi-million arms deals has been mixed.