The lease and leaseback agreement includes a 49% stake of newly formed Aramco Oil Pipelines Co and rights to 25 years of tariff payments for oil carried on Aramco's pipelines, it said in a statement.
Aramco will retain a 51% stake in the new company.
EIG, which has invested more than $34 billion in energy and energy infrastructure, was the deal's underwriter and will work with Aramco in the coming days to decide on other parties for the consortium, a source familiar with the deal said.
Abu Dhabi state investor Mubadala is in discussions on being part of it, a spokesman said.
Aramco will retain operational control of the pipeline network and assume all operating and capital expense risk, the companies said. The deal will have no impact on Aramco's oil production.
Aramco will also offer so-called "staple financing" which the buyers can use to back their purchase, sources have told Reuters.