Picture this: Mean Joe Greene has had a tough day. He's tired, he's sweaty. A young boy intercepts him on his way to the locker room. "Gee, Mr. Greene," he says, "want some of my . . . Perrier?"

Greene takes the dainty pear-shaped green bottle and guzzles all 6 1/2 ounces in a mighty swig. "Thanks, kid," he says, "but where's the lime?"

Bruce S. Nevins, the president of the Great Waters of France, a subsidiary of Perrier, liked the idea. "That makes a heck of a lot of sense," he said. "All the Dallas Cowboys drink Perrier. Maybe he could give it to Roger Staubach."

"Nevins and Perrier have done a brilliant job associating with running and sports," said Dick Auletta, whose firm handles public relations for Dannon Yogurt, which along with Perrier, is one of the biggest corporate sponsors of road racing today.

In the mid-1970s, Nevins sold Levis to the Chinese. In 1977, he began selling fizzy water to runners. Ron Davis, vice president of marketing services, said, "Today runners drink seven times as much Perrier as nonrunners," which is quite an accomplishment when you consider that water fountains are still free.

Perrier will sponsor a series of seven road races in 1980, including Sunday's Perrier Cherry Blossom 10-miler in Washington and the New York Marathon.

Perrier spent over $100,000 on last year's Marathon, according to Nevins. The company will spend approximately $500,000 on running in 1980, including $20,000 on Sunday's race, said Jeff Darman, Cherry Blossom spokesman.

Darman, former race coordinator, said that approximately 10 runners are receiving corporate sponsorship this year. "Not one penny of our budget is going to travel expenses," he said, "and we're not charging an entry fee either. How they come is their own business."

The best thing about Perrier, said Darman, is "they leave the technical things to us. They're knowledgeable about running, but they don't make silly, outrageous demands."

But, Darman added, "they haven't gotten into making a statement the way Dannon has." The Dannon Road-Runners Club of America Series has a ceiling of $1.50 on entry fees.

Sources say that Bill Rodgers, the defending champion and the only runner in the race sponsored by Perrier, will receive $5,000 from the company. Rodgers said, "I have no contract or arrangement with Perrier. At one time, I had made a plan to run a certain number of their races. This year, it is totally spur of the moment."

And what does Perrier get for its money?Product identification. Market positioning.

"It's a subliminal return on your investment," said Auletta. "It's difficult to quantify."

Perrier will not release sales figures. But Barbara Long, director of marketing services, said that when Perrier was introduced here in 1976, "our product awareness in Los Angeles was very low. Today, in Los Angeles, New York, Washington, San Francisco, it's over 90 percent. At least 25 percent of that is from running."

Auletta points to more graphic evidence. "If you go into a class restaurant and order a Perrier, it's delivered in a bottle. You don't see Canada Dry delivered in the bottle."

And in even classier restaurants, the bottle is used as a vase.

In 1976, Perrier was a gourmet store staple, definitely upper crust. Nevins, looking for a way to "make the transition" to being a mass-market item -- "a healthful alternative beverage," seized upon the healthful, alternative sport -- running.

Perrier sponsored its first race in August 1977, the 7.1-mile Falmouth road race in Massachusetts. Runners straggling across the finish line provided a captive test sample for the drink, given out free. Some who did not realize the drink was carbonated, got sick to their stomachs and spit it out. Rodgers, who drank his first Perrier that day, said, "I wasn't quite as fond of it as I am now."

Americans, including Rodgers, soon got used to it (the company estimates about 50 percent of all adult Americans have tried it). "Why don't they just walk up to a water fountain?" said Long. "Americans are used to carbonation. wWe're the Pepsi generation. We're used to bubbles."

Perrier has cut back somewhat on the number of races its sponsors, although it has not reduced its racing budget. It is not anxious to be seen as the in drink for the in crowd, jumping to the in sport. "That's what we're accused of," said Sally Nesstler, national special events coordinator.

The accusation arose when Perrier decided to cosponsor a 19-tournament racquetball tour with Ektelon. Although Perrier does not spend as much money on racquetball, and Nevins doubts that it will, he said, "The running boom has begun to level out. Most people have already decided whether they're going to be runners. People are going to begin to look for other things.

"It's anticipating (the racquetball boom)," Nevins said, "but also preempting it. We're getting in on the ground floor, where it doesn't cost millions of dollars. We're not Pepsi. We have to be a little more visionary.'