The plan to float about 5 percent of Aramco aims to raise $100 billion (£74.9 billion) and is a centrepiece of Vision 2030, a programme to diversify Saudi Arabia's economy away from oil that is championed by Crown Prince Mohammad bin Salman.
Saudi Aramco declined to comment.
The existing banks working as advisers - JPMorgan, Morgan Stanley and HSBC - are seen as front-runners for the global coordinator roles, with the possible addition of a small number of other banks, a third source said.
The source said the final number of banks with the mandate was still uncertain.
The trio joined Moelis & Co and Evercore, who had already been appointed as independent financial advisers.
Wall Street firms Goldman Sachs and Citi are among the many banks vying for a role, sources said.
JPMorgan, Goldman, Morgan Stanley and HSBC declined to comment. Other banks were not immediately available to comment.
Saudi officials have said Aramco will be listed in Riyadh and possibly one or more international exchanges.
Aramco has said it is looking at New York, London, Tokyo and Hong Kong as potential venues for the partial listing of the state-run firm.
(Additional reporting by Hadeel Al Sayegh; Editing by Edmund Blair)
By Dasha Afanasieva, Saeed Azhar and Tom Arnold