The unemployment rate in the District of Columbia climbed to 10.5 percent in March, up 0.4 percent from February, while joblessness in the metropolitan area remained at the record 6.1 percent level first reported in February, city officials said yesterday.

The city figures, officials said, indicated a worsening trend that could lead to record levels of unemployment in Washington by early summer.

District officials, including Mayor Marion Barry, blamed federal budget cutbacks and the continuing sluggishness in the national economy for the steadily climbing jobless level.

The March report said 32,500 persons were out of work in the District in March compared to 31,200 in February and 26,900 in March 1981, when the unemployment rate was 8.6 percent.

Lawrence Thurston, a labor economist for the District's Department of Employment Services, said the District figures released yesterday represented a marginal change in the month-to-month reporting.

But he warned that the upward trend means the city probably will soon exceed the record 10.9 percent rate reported in June 1977. "I do expect the rate to go higher with the traditional influx of students and in some cases parents" looking for work in the summer, Thurston said.

The city reported a slight increase of available jobs, up 500 from 598,900 in February to 599,400 in March. District residents hold 276,000 of the jobs within the city, officials said, compared to a year ago when District residents held 285,400 jobs, or 9,200 more positions. In the metropolitan area as a whole, 1,583,700 persons are employed compared to March 1981 when 1,593,100 persons were employed, a net decrease of 9,400 jobs.

Employment Services director Ivanhoe Donaldson said the city has stepped up its employment counseling and job location services to deal with the rising number of jobless residents.

"Potentially, we could have been worse off if District government resources had not been there to cushion" the loss of federal job training funds and other program assistance, Donaldson said.

Thurston also cautioned that the .4 percent change reported in local unemployment figures was not comparable to reporting changes in the national figures where he said changes of as little as .2 percent are considered more significant. He said the District and most states use a less sophisticated method of compiling data and making seasonal adjustments.

The change "has to be .8 to 1.2 percent before you talk about significant change" in the monthly figures, Thurston said.