The Federal Aviation Administration and the Metropolitan Washington Council of Governments agreed yesterday to see if they can reduce jet airplane noise in southern Fairfax and Prince George's counties by changing the routes of airliners leaving National Airport to the south.

At the same time, the COG board decided not to tamper with existing departure routes to the north, up the Potomac River, until further study.

The joint decision to conduct a test on southbound departure routes was the latest chapter in attempts to solve one of the area's longest-running insoluble problems: noise from planes using close-in National Airport, which is popular with members of Congress.

What the FAA will do, starting next spring or early summer, is instruct jet airline pilots taking off that they must fly 10 miles south before turning toward their destinations.

At the moment, jets fly five miles to the south before turning at a point near the Woodrow Wilson Bridge. By adding five miles to that, planes would stay above the Potomac River longer and would climb higher before turning over the residential subdivisions of Fairfax and Prince George's counties.

The FAA and COG will conduct about a three-month test and compare noise measurements and citizens complaints before and after the test. Then a decision will be made on which procedure to use.

The five-mile extension was proposed by COG to the FAA for both northbound and southbound departures in an earlier chapter of the controversy.

On northbound departures, jet planes are supposed to fly up the Potomac River for about 10 miles, to the Cabin John area, then apply full power and turn toward their destinations.

The FAA said it could add only three miles to that departure program, not five, because five miles would place National Airport planes in the zone belonging to air traffic controllers at Baltimore-Washington International Airport. A three-mile addition would mean all planes would turn over the community of Potomac.

The COG staff conceded that "less people would be impacted than if the status quo condition prevailed." However, it noted, development plans in the Potomac area call for heavier densities in the future. Furthermore, the staff said, a three-mile extension would guarantee that all planes would pass over Cabin John en route to Potomac. At present, about 20 percent of the planes turn off before reaching Cabin John.

That is because, as the FAA concedes in private, many planes start turning before they reach 10 miles and they float over such areas as McLean and Falls Church.

The real difficulty of the National Airport fracas, which is that somebody loses every time somebody else gains, came bursting to the fore when Fairfax County Supervisor Martha V. Pennino suggested more study of the northbound departure problem.

Rockville Mayor William E. Hanna Jr. seconded Pennino, but only after getting the motion changed to preclude a study of anything that would change existing flight patterns over Maryland.

"That's because that one arrow (on the map) points at Rockville," Prince George's County Councilman Francis B. Francois observed to some laughter.

The COG board yesterday also forwarded for local government comments the proposed fiscal 1980 COG budget of $6.4 million. Most of that money will come from federal sources. Local governments would contribute $921,075. The proposed budget represents a 4.7 percent increase.