Greenberg is one of the crowd, his shirt sleeves rolled up, working the telephone. It takes a while to spot him. Then you notice the slightly raised perch, the dense traffic around one desk, the husky, bald man seated there. This is the post from which Greenberg runs the huge firm, dispensing decisions in his gruff, rapid-fire style.

Meanwhile, he fields phone calls from some of the nation's wealthiest investors. His list of clients includes such takeover artists as Irwin L. Jacobs, and Kohlberg, Kravis, Roberts & Company. After 40 years in the business, much of it spent buying and selling stocks of companies involved in takeover situations, Greenberg has developed a vast network of market intelligence and an instinctive feel for how much of a company's stock can be bought at a certain price level without causing a ripple.

Moreover, he's not shy about voicing his own opinion. ''Alan has talked me out of buying lots of stocks that he didn't like,'' says Donald Trump. On the other hand, it was with Greenberg's encouragement that the real estate developer took a big position in Federated Department Stores in 1987. He bought in at $29 a share and more than doubled his money when the Campeau Corporation acquired all the company's stock and added Federated to its North American retailing empire.

For a client like Donald Trump, one of Greenberg's most admirable traits is his ability to keep a secret. ''If Alan represents me on something,'' says Trump, ''nobody else is going to know about it. He's extremely closemouthed.'' Within the firm, a Greenberg client is identified as a numbered account. Only two other people know what buy and sell orders Greenberg has placed for whom: an assistant who sits beside him and the head of the firm's compliance department. After receiving the confirmation tickets, Greenberg personally puts them in envelopes, seals and addresses them and places them in the outgoing-mail basket.

In camouflaging his activities, Greenberg has a crucial advantage. Bear, Stearns clears trades for about 800 regional brokerage houses, arbitrageurs and money managers. As a result, the Bear, as the firm is known on Wall Street, currently clears a remarkable 8 percent of all trades done on the floor of the New York Stock Exchange, constantly buying and selling large blocks of stock. ''There's nothing unusual about us placing big orders, so our tracks are always covered,'' says Greenberg. ''Nobody knew,'' he adds with evident pride, ''what we were doing when we bought huge positions in Kroger for K.K.R.'' - a reference to Kohlberg, Kravis's attempted takeover of the supermarket chain last year. ''Everybody was guessing and they weren't even close.''

Despite Greenberg's close association with so many takeover specialists, he has not figured in any insider-trading cases. The firm has been involved with some clients that subsequently became embroiled in legal troubles, including Wedtech and the Belzberg family of Canada. But Bear, Stearns has so far emerged unscathed.

Greenberg makes no bones about the fact that Ivan Boesky, who pleaded guilty to a charge related to insider trading, used to call him frequently. ''We frankly had the opportunity to make some of the mistakes that other people made,'' Greenberg says. ''He asked me to do some things, and I said, 'It can't be done, Ivan,' just like that.'' Robert M. Steinberg, head of Bear, Stearns's risk-arbitrage department, remembers one call during which Boesky had evidently asked Greenberg a question. ''Alan said to him on the phone, 'Ivan, can you keep a secret?' And I can just anticipate what Ivan must have been thinking at that moment: 'I got him! I got him!' '' At that point, says Steinberg, Greenberg told Boesky, ''Well, so can I,'' and hung up with a roar of laughter.