Days after a BlackRock Inc.-led group signed an $11 billion lease agreement for some of Saudi Aramco’s natural gas facilities, the energy giant was inundated with calls from funds around the world eager for a slice of the business.

Emboldened by that demand and driven by a desire to shore up the balance sheet, Aramco executives in Dhahran have lined up the most ambitious privatization plan in the company’s 93-year history. In the months since that deal with BlackRock’s Global Infrastructure Partners, the firm has pushed ahead with a wave of divestments spanning everything from energy facilities and even real estate.