NATHAN'S HUNT FOR AN UPTURN

Almost every day at rush hour this summer, Nathan's Famous deploy edsquadrons of green-and-yellow uniformed teen- agers in Times Square tohand out coupons and promote the New York hot -dog institution's 65th birthday celebration. But at summer's end, Mu rray Handwerker, the sonof Nathan, the restaurant's founder, stood up before angry stockholders at the company's annual meeting and could think of nothing celebratory to say.

For New York appears to have lost some of its ardor for Nathan's legendary hot dogs - once a nickel and now 85 cents - and its delicatessen line.

The company has had to shut down 23 restaurants in the metropolitan area in the last year. It now operates 13 and franchises 10 others in the metropolitan area, and is seeking to develop a new market in the Sun Belt.

7 Units Sold to Marriott

Nathan's sold seven metropolitan units to the Marriott Corporation last fall, and at the annual meeting, there were reports that Marriott was considering acquiring Nathan's. Mr. Handw erker vigorously de nied it, but said that ''a major party is interested in discussions'' on acquiring the company and that talks ''are taking place on a se rious level.''

Michael Rouche, spokesman for Marriott, said he would not discuss the possibility of an acquisition. ''The only thing I can say is we are always looking for new locations and New York is very attractive,'' he said.

In the last year, Nathan's has been forced ''to bite the bullet,'' said Murray Handwerker, president of the company, which started with a stand in Coney Island.