Institutional bids received during the first 12 days of the book-building period, which continues until December 4, now stand at more than 118 billion riyals (US$31.5 billion, €28.6 billion), Aramco said.
The energy giant is offering 1.5 percent, or three billion, of the company's shares on the Saudi Stock Exchange (Tadawul) at an indicative price range of SR30-SR32 ($8-$8.53).
Kuwait's government will invest as much as $1bn in the IPO as the kingdom asks regional allies to bolster the record share sale.
Aramco will list 1.5 percent of its shares this week, targeting local investors, both large and retail.
Meanwhile, Saudi Arabia's Tadawul has introduced an equity index cap of 15% which is set to address concerns over the weighting oil giant Saudi Aramco will have when it lists on the exchange.
Overall demand so far totals 1.7 times the amount of shares on offer, with institutional investors having until Dec 4 to put their bids in.
The prevailing argument is that Saudi Arabia wants to deeper cuts in order to boost the value of its long-awaited IPO of Saudi Aramco: according to Reuters, one source told the news agency that "They [the Saudis] want to surprise the market", while another two sources said the latest OPEC analysis showed a large oversupply and build up in inventories in the first half of 2020, if no additional cuts were made.
"Each index constituent weight reaches or exceeds the threshold will be limited in accordance with defined borders", Tadawul said in a statement on Monday.
This agreement is the centerpiece of Crown Prince Mohammed bin Salman's plans to diversify the Saudi economy away from oil.
The fund has interests in ports, airports and power distribution globally.
The current deal runs to March.
As Opec's de facto leader, Saudi Arabia is expected to use its position to push other members to tighten their compliance with the group's agreed oil production limits, while cutting its own output even further than it needs to.