Then on Sept. 14, on its path to going public, Aramco was jarred by an aerial attack on its production facilities, blamed on Iran, that temporarily cut its oil output in half. The attack underscored the risk of operating in the Middle East, but it did not deter the march to an I.P.O.

Deal makers soon fanned out over Asia, Europe and North America to gauge interest in Aramco by Fidelity Investments, Capital Group, BlackRock and other major investors. To make Aramco more attractive, the banks persuaded it to establish an enormous investor dividend, or annual payout — $75 billion a year.

But in meetings with roughly 80 mutual funds, hedge funds and sovereign wealth funds, underwriters and investors said, potential buyers balked at the $2 trillion valuation, which struck them as too high relative to other major oil companies and in light of low oil prices, climate-change concerns and other geopolitical pressures.

“We felt that a valuation in the range of $1.2 to $1.3 trillion would represent fair value,” or a reasonable price, “but it would need to I.P.O. at less than that to offer decent upside,” or investor profit potential, said Tal Lomnitzer, a portfolio manager at the fund company Janus Henderson who participated in the early investor discussions.

His was in some ways the typical buyer’s position at the onset of a negotiation: to argue for the lowest price in hopes of making money on the purchase if Aramco shares went up in public-market trading. But given the wide gap between views like Mr. Lomnitzer’s and the Saudi government’s $2 trillion expectations, some of the bankers were concerned.

Then came the meeting on Oct. 15 at Aramco’s headquarters in Dhahran on the kingdom’s Persian Gulf coast, and one that would follow the next day. Of all the crucial moments in the lead-up to the I.P.O., these gatherings may have been the most tense, according to four people who either attended the meetings or were briefed afterward. It was then that some of the bankers — motivated by the promise of enormous fees for evaluating the oil company’s investment potential and then selling shares to respected investors — clashed with kingdom officials and other advisers who were fixating on an increasingly elusive $2 trillion deal.

The banks, who had been sizing up investor demand for the I.P.O., delivered their findings to Amin H. Nasser, Aramco’s chief executive. Mr. Nasser was angry and taken aback by the news, said two people who were in the room and three others briefed on it later. He pointed out that some of the bankers had promised an Aramco valuation of even more than $2 trillion, and that his company had curbed spending plans and made other changes to accommodate the $75 billion dividend.