Mr Macron’s demand came shortly before Theresa May held face-to-face talks with the crown prince in which she promised to send a “robust” message about the murder, as well as the war in Yemen.
Mohammed has appeared isolated among fellow world leaders at the G20 summit in Argentina, with the exception of a vigorous public handshake from Russia’s Vladimir Putin - himself treated as a pariah by some of those present.
The crown prince was filmed in an apparently tense conversation with Mr Macron, telling the French president not to worry.
Mr Macron responded: “I am worried...You never listen to me.”
The widely-shared video showed bin Salman replying: “I will listen, of course.” It concluded with Mr Macron saying: “I am a man of my word.”
Later, an Elysee Palace official said the president had conveyed a “very firm” message on the need for international involvement in the Khashoggi investigation, currently the subject of two separate probes by Saudi Arabia and Turkey.
Mrs May said she would use her meeting to press Britain’s call for a credible and transparent investigation into Mr Khashoggi’s death in the Saudi consulate in Istanbul in October and urge Mohammed to ensure his country offers full co-operation to the Turkish investigation.
Mrs May was also pressing the Crown Prince to ensure his country’s engagement in talks in Stockholm next week to find a political solution to the bloodshed in Yemen, where a Saudi-led coalition is fighting Houthi rebels.
Following the meeting, a Downing Street spokesman said: “The Prime Minister stressed the importance of ensuring that those responsible for the appalling murder of Jamal Khashoggi are held to account, and that Saudi Arabia takes action to build confidence that such a deplorable incident could not happen again.
“Noting the steps taken by the Saudi investigation since the Foreign Secretary had met with the crown prince and King Salman on 12 November, she encouraged the crown prince to ensure that Saudi Arabia co-operated fully with the Turkish authorities and worked to bring both investigations to an acceptable close.
“To ensure full accountability, there needed to be full transparency about exactly what had happened and who was responsible, in line with the commitments made by King Salman when she spoke to him on October 24.”
On Yemen, the spokesman said that Mrs May set out the “urgent need” to bring an end to the conflict and bring relief to millions threatened by famine.
“The Prime Minister stressed that the humanitarian situation remained dire and reaffirmed UK commitment to making progress on improving the situation, including through a UN Security Council Resolution,” the spokesman said.
“The Prime Minister reiterated UK support for Saudi Arabia’s security, including the very real threats posed by Iranian interference in Yemen.”
Amnesty International UK’s director Kate Allen said Mrs May should “politely but firmly insist that only a UN investigation into the grisly murder of Jamal Khashoggi is going to be sufficient”.
Speaking to Sky News ahead of the meeting, Mrs May said: “It is the relationship we have with Saudi Arabia that enables me to sit down with him and be robust on our views on two issues”.
On the Khashoggi case, she said Riyadh must “ensure that their investigation is a full investigation, that it is credible, that it is transparent and that people can have confidence in the outcome of it and that those responsible are held to account”.
And she said she was “very concerned” about the humanitarian situation in Yemen resulting from the long-running conflict.
“Now is the time, there is an opportunity to find a solution,” said the PM.
“I will be encouraging all parties, including the Saudi Arabians, to ensure that they sit down at the upcoming UN-led talks in Stockholm in the coming days and weeks and find a way through this, find a political solution for a stable future for Yemen.”
Mrs May also held talks with the summit’s host, President Mauricio Macri, at which the long-standing dispute over the sovereignty of the Falklands took a back seat as discussions focused on trade.
Mrs May presented keen Queen fan Macri with a book about the rock band, while the Argentine president gave her a special cup for drinking national beverage Yerba Mate herbal tea.
Mrs May is the first Conservative Prime Minister to visit Argentina since the 1982 war, and only the second serving PM to do so, after Tony Blair briefly crossed the border from Brazil in 2001.
Mr Macri has toned down official rhetoric over the islands Argentina knows as Las Malvinas since taking over from Cristina Kirchner in 2015.
This week’s announcement of a new air link between the Falklands and the South American mainland via Argentina has been greeted in London as an indication that relations are headed in a positive direction.
Speaking during her flight to Buenos Aires, Mrs May said the establishment of the flight showed “a different relationship developing”.
“I am talking to President Macri about issues about trade and opportunities for trade, but our position on the sovereignty of the Falklands hasn’t changed and will not change,” she added.
The PM is using the G20 gathering to push her “global Britain” message, telling fellow leaders that her Brexit deal will be good for the world economy.
On Saturday, she will hold bilateral talks with leaders of Canada, Australia, Chile and Japan to discuss opportunities for future trade links, as well as discussing the progress of the Turkish murder probe with President Recep Tayyip Erdogan.