The National Aeronautics and Space Administration backed down a little Tuesday from its plan to build its own plant to manufacture advanced solid rocket boosters for the Space Shuttle.

But NASA made it clear it would like to do just that, a step that would take booster production out of Utah.NASA briefed reporters on its plans, which apparently have been trimmed at least a bit by the Office of Management and Budget at the White House. OMB has been under pressure from Sens. Jake Garn and Orrin Hatch, both R-Utah, to spike NASA's plans for its own rocket factory as contrary to the Reagan administration's "privatization" policy.

Adm. Richard Truly, associate NASA administrator, testified two weeks ago that the agency had decided not to go to the Morton-Thiokol company's Brigham City plant for the Advanced Solid Rocket Motor. Truly added that he thought OMB objections would be quickly overcome.

Garn and Hatch apparently had more clout than NASA officials thought, although from this week's briefing, the agency intends to hold a competition for manufacture of the booster that may be hard for Morton Thiokol to win.

NASA would like to build the big boosters at a location nearer to Cape Canaveral than northern Utah. It has identified three potential sites: Cape Canaveral itself, a former Tennessee Valley Authority tract at Yellow Creek, Miss., and its own National Space Laboratory, also in Mississippi. The Mississippi sites are on waterways that could accommodate barges capable of carrying booster rockets, something Brigham City can't match.

The Advanced Solid Rocket Motor, as described by Truly and by NASA Project Manager Lowell Zoller, would have joints designed to close under the pressure of firing, not open like the current rocket motors. It would be produced under more controlled conditions, computer-directed, to make the boosters more uniform and presumably safer. NASA's cost could hit $1.2 billion.

NASA's plan calls for bids on building a government-owned, contractor-operated facility, which would eliminate Thiokol. But it permits alternatives, including bids for production at a contractor-owned plant, such as Thiokol's.