NEW YORK, JUNE 14 -- The New York Stock Exchange, eager to stay abreast of high-tech rivals but reluctant to offend the traders who crowd its floor each day, has cautiously dipped its toe to test the unknown waters of nighttime, computerized stock trading.

The exchange confirmed today that it hoped to gradually introduce after-hours trading in a series of steps beginning later this year, with the goal of achieving global, round-the-clock operations by the year 2000.

The additional trading, which would first appear in a limited form, would take place on institutional investors' computer screens after the 4 p.m. closing bell clears the exchange itself.

If interest is strong enough the exchange next year would set up a more elaborate system of trading in which stocks would be bought and sold at three separate auctions during the night. While most U.S. investors would be asleep, the exchange could recapture significant amounts of after-hours trading business that has migrated to exchanges overseas.

With some form of round-the-clock trading, big American stock buyers such as pension funds or mutual funds would have the chance to respond immediately to news or other developments that affect stock prices, without having to wait until the exchange opens at 9:30 a.m. In theory, a small investor who had a great idea for a stock buy at 2 a.m. also could arrange for a broker to make the purchase at 5 a.m., although few expect this to be a large part of the expanded business.

''We want to do it, but not if it becomes too expensive,'' NYSE Chairman John J. Phelan Jr., told a news conference in Nashville, after meeting with officials from regional brokerage J.C. Bradford & Co.

Phelan said he is meeting with member firms around the country and will know by mid-September whether they approve of after-hours trading.

The New York financial community generally welcomed the proposal. Investment firms said it was important not to cede round-the-clock stock trading to foreign markets, especially the automated one in London where much off-hours business now takes place.

"I would think everybody would be in favor {of off-hours trading}, because it would mean more business. ... More volume makes more money," said Brad Weeks, a senior vice president and head of sales trading at Donaldson, Lufkin & Jenrette, reflecting the view of an industry suffering from the recent cutback in trading activity.

Stanley Shopkorn, vice chairman of Salomon Brothers Inc., issued a statement saying: "The auction market's change to 24-hour trading will make the NYSE more effective in the years to come. This extension of trading hours is a result of globalization."

But many details of the off-hours trading plan remain to be decided, and the proposal could arouse controversy in some quarters. In particular, the plan could face opposition from stock specialists who now manage the trading for each stock in person on the exchange floor and who might be bypassed by a computerized system.

''It's an incredible surprise,'' one veteran specialist told the Associated Press. "The incredible thing is that the floor hasn't lynched anyone yet.''

But Robert W. Seijas, a specialist at the firm of Lasker, Stone & Stern who handles trading in Cooper Industries Inc. and other issues, dismissed predictions that the introduction of computerized trading would erode the role of the specialists. He predicted demand for off-hours trading would be limited, partly because many big investment houses did not want to pay for people to stay up all night working.

In its initial phase, the exchange proposes to allow after-hours trading of large blocks of stock at the price at which the stock closed at the end of normal trading hours.

A much bigger step toward an off-hours market could come next year, when prices would be allowed to fluctuate between closing and opening, as stocks were sold at auctions at 8 p.m., midnight and 5 a.m.

Phelan said that in subsequent phases the market would conduct ''some sort of rudimentary trading during the night'' before achieving bona fide round-the-clock trading.