The U.S. Army School of the Americas in the old Panama Canal Zone has graduated more than 42,000 Latin American soldiers from its various courses over the last 37 years.

The Pentagon likes to portray it as a haven for mutual understanding, a place to develop what the official briefing book calls "professionalism" and "effectiveness" and "military-to-military relationships" in the region. Critics, especially on the left, look on it as a school for dictators and scoundrels.

Leopoldo Galtieri once attended classes in its nine tranquil, tropical buildings before going on to become the Argentine general and president who invaded the Falkland Islands. Gen. Gustavo Alvarez, the controversial commander of the Honduran armed forces, is also an alumnus. Before the 1979 overthrow of Nicaraguan strongman Anastasio Somoza, 4,693 members of his widely despised National Guard went through the school. These figures and those on the chart accompanying this article give the number of Latin Americans trained there between the school's founding in 1946 and April 20 of this year.

Now the institution has a key role in efforts to reshape and expand the armies Washington underwrites in fighting the growing wars of Central America. Enrollment in the entirely U.S.-funded school has surged from a low of about 700 in 1980 to a projected attendance of more than 2,400 soldiers, cadets and officers this year. Almost half are Salvadoran.

Just as the school has taken on this special importance its future in Panama has been thrown into doubt by a clause left dangling in the 1977 treaty ending exclusive U.S. control of the Panama canal. One of the agreements attached to the accords stipulates that the authority of the United States to school Latin American military personnel here will expire on Oct. 1, 1984, unless Panama and Washington agree otherwise.

The late Panamanian leader, Gen. Omar Torrijos, who negotiated the treaties, used to say he would turn the school into a "kindergarten" when the time came. Panamanian officials today say that some high schools in nearby Colon have actually put in requests to the government for the buildings.

Until this month, the issue of the School of the Americas' future was sidestepped by both the Pentagon and the Panamanians. Now discussions have begun, and Panama's leaders, with a strong sense that time is running out for Washington, are looking to drive a hard bargain.

As they put it, the presence of the school is potentially "a spark" that could touch off the political passions always kindled here by the issue of the canal and the American presence.

The specter of leftist unrest, particularly, is raised as a real fear. But in this country where much of the left has been co-opted by the government, it is perhaps also a threat.

The School of the Americas is "more than an installation, more than a school, it is something almost romantic for the left," said one Panamanian military official concerned with the negotiations. "They say they make dictators there, they make repression there."

But one of the more prominent leftist organizers in Panama City said last week that for the moment at least, given the problems of unemployment and the debate surrounding a recent constitutional referendum, "almost no one is talking about the training."

The presidential elections planned for next year, the first in Panama since 1968, complicate matters even further.

Gen. Ruben Dario Paredes, a School of the Americas' alumnus, is the current commander of the Panamanian National Guard and, as such, the current government strong man. He is also likely to be the leading presidential candidate in 1984.

His position on the school, stated in an interview last week, is that the buildings where it is housed at Fort Gulick will revert to Panama and fly the Panamanian flag. "This condition is not negotiable," he said.

"It is possible that Gulick will continue as a school, but it depends on the amount of good will shown by the United States," said Paredes. "If we don't come to an agreement, they can move the school to Florida, Puerto Rico or Honduras. This is a very sensitive issue for Panamanians. If the school stays, the United States has to make important concessions for people to accept it here."

Lt. Gen. Wallace H. Nutting, head of the United States Southern Command in Panama, said that Washington "and all of Latin America" wants to keep the school in Panama. Moving it would be an expensive and clumsy process at a crucial time and putting it in politically vulnerable Honduras is not likely despite an invitation from Gen. Alvarez to do so. Sites there have been looked at but "not seriously," according to Nutting.

The American general declined to comment on specific negotiating points except to say that "symbols obviously are important to Panama and the United States understands that."

Other U.S. officials noted, however, that Panama reaps at least two obvious advantages from the school's presence: the ease and relative cheapness of training its own soldiers there (almost 4,000 since the school opened), and an estimated $12 million to $15 million the school and its students put into the severely depressed economy of Colon, the nearby Atlantic port.

According to sources involved with the negotiations, they are being led by Panamanian Lt. Col. Julio Ow Young and the school's director, Col. Nicholas A. Andreacchio.

In sessions on May 5, 10 and 12 the Panamanians at first pushed for a Panamanian director or codirector of the school, but have gradually backed off that position since it was made clear to them the U.S. Congress would be unlikely to approve complete funding of what would amount to another country's international military academy. Now there is discussion of a more nebulous "council" including Panamanian and other Latin American officers, who would periodically "evaluate" the performance of the school.

"We have to look for a solution that saves face for Panama," explained one senior Panamanian official.