A White House committee has decided the nation's $6 billion nuclear weapons-building program should be run by a quasi-independent agency attached to the Commerce Department, according to informed congressional sources.

The recommendation, sources said, was agreed upon Tuesday by a Cabinet council studying how to dismantle the Department of Energy, which runs the weapons program.

The recommendation must be approved by the president before being sent to Congress. Reagan has announced that he will follow through on his campaign pledge and recommend next year doing away with DOE. The decision to put nuclear weapons under the commerce secretary is expected to run into serious trouble on Capitol Hill, sources said, particularly with members of the Senate and House armed services committees.

"The members feel so strongly," one Senate source said, "that dismantling of DOE might not happen." Senate approval is needed for any Reagan reorganization program.

A letter sent to the president last week, signed by 13 senators, including Senate Armed Services Chairman John G. Tower (R-Tex.), said "we would strongly object" to moving the nuclear weapons program to the Commerce Department. Instead, the letter suggested a "return to an agency much like the former Atomic Energy Commission" or the establishment of "a semi-independent agency within the Defense Department."

The program has been under civilian departments with other major interests the past seven years. As a result, according to Capitol Hill sources, the senators say they believe the complex of seven government-owned plants that build all the country's nuclear weapons has been allowed to run down just as the biggest warhead production in history is getting under way.

When the old Atomic Energy Commission was disbanded in 1974, the weapons program was made part of the Energy Research and Development Agency (ERDA). When ERDA was absorbed by the Department of Energy, the weapons program went along.

Over those years, as energy became a more important domestic issue, attention of the top DOE officials focused primarily on it, rather than on weapons building. The last two energy secretaries "knew little about what was going on," one nuclear scientist in the program said yesterday.

The White House committee, according to sources, considered three proposals for the weapons-building program: attaching it to the Interior Department, a move favored by Energy Secretary James B. Edwards; making it independent, a solution supported by Defense Department officials, and making it semi-autonomous but under the commerce secretary, which reportedly was favored by Office of Management and Budget officials.

A major factor weighed against putting the program under the Defense Department, sources said. Such a move would violate a standard set during the first debates on the Atomic Energy Act in 1946 that the building of weapons should be controlled by civilians, not the military.

An argument working against the creation of an independent agency was the Reagan administration pledge not to create new federal agencies.

Sources said that transferring the program to Interior was ruled out partly because of the controversial character of Interior Secretary James G. Watt and partly because Interior has, by tradition, an environmentalist tilt, just the opposite tilt wanted for the weapons-building program.

Commerce had the advantage of having some high-technology programs and other quasi-independent agencies, such as the Federal Aviation Agency and the National Oceanic and Atmospheric Agency. In addition, Commerce Secretary Malcolm Baldrige is highly respected within the White House.

The weapons-building program consists of the two national nuclear laboratories at Los Alamos, N.M., and Livermore, Calif., plus the Sandia Laboratories, the nuclear test site in Nevada, nuclear material production reactors at Savannah River, S.C., Oak Ridge, Tenn., and Richland, Wash., and factories and weapons-assembly facilities at five sites across the country.