Five House Republicans are strongly urging President Reagan to cut more than $20 billion from the 1982 Pentagon budget in an effort to step up pressure on the White House to force Defense Secretary Caspar W. Weinberger to accept spending reductions.

The House members, associated with the "gypsy moth" group of Republicans from Northeast and Midwest districts, have told Reagan in a letter that such a cut would not harm the nation's defense but that it is "necessary for your overall economic program to succeed."

The letter predicts that "there is certain to be a public backlash against a military buildup that worsens the economy, is financed through cuts in vital social programs, and appears to be a reckless spending spree."

The gypsy moths are mainly from urban areas and number about two dozen, grouped in an informal caucus. They are considered by Democratic strategists to be vulnerable in next year's midterm elections, mainly because of redistricting proposals: a position they are using to give them leverage with the Republican leadership.

In the letter to Reagan, the five ask for defense spending to be subjected quickly to the same rigorous controls applied to domestic programs, and say: "It has become painfully evident that without such restraints the large deficit that will occur will cause federal 'crowding out' of the private capital markets with resultant high interest rates which deny us the economic growth your tax program seeks."

The signatories are: Reps. S. William Green (N.Y.), Harold C. Hollenbeck (N.J.), Margaret M. Heckler (Mass.), Thomas J. Tauke (Iowa) and Rep. Arlen Erdahl (Minn.). They have contacted others associated with the group to ask for support and expect to receive it when the House returns from recess next week.

Staff in the offices of a number of "gypsy moths" -- the name emerged following the publicity given to the Democratic "boll weevils" who broke with their party to vote for the Reagan budget -- indicated that they would expect their members to express support.

But others are less enthusiastic about the letter, drafted in Green's office. Rep. Jim Dunn (Mich.), for example, said he believed that such proposals for scaled-down defense spending increases should come from the armed services committee rather than the gypsy moths.

The letter says there is room for trimming defense increases by funding only programs that are "truly needed." It suggests that the five planned methods of strategic nuclear response could be reduced to three without damaging national security.