Senate Minority Leader Robert C. Byrd (D-W.Va.) urged President Reagan yesterday not to name William Bradford Reynolds as associate attorney general during Congress' August recess, saying such a move "would be an insult to the Senate and an affront to the Constitution."

In a letter to Reagan on behalf of all 47 Democratic senators, Byrd reminded the president that the Senate Judiciary Committee rejected Reynolds for the Justice Department's No. 3 position last month.

He said that "a recess appointment of the same individual to the same position would be inappropriate and unacceptable."

Even Republicans cautioned against a recess appointment. Senate Majority Whip Alan K. Simpson (R-Wyo.) said he did not want to see "a subterranean campaign of some kind that would be a distraction from the heavy work load we have around here."

The swift reaction appeared to let much of the steam out of efforts to revive the Reynolds nomination. However, Simpson and Majority Leader Robert J. Dole (Kan.) said they would support a "discharge petition" to force the nomination to the floor.

Dole said there was "widespread" GOP support for that idea.

Reynolds, who has headed the Justice Department's Civil Rights Division for the last four years, was voted down 10 to 8 amid criticism that he had been lax in enforcing civil rights laws and had misled the Senate Judiciary Committee in sworn testimony.

Republicans Arlen Specter (Pa.) and Charles McC. Mathias Jr. (Md.) joined the eight committee Democrats in opposition.

Byrd's letter to Reagan cited a report in yesterday's Washington Post which said that White House officials are calling senators to gauge whether they have enough support to force the Senate to vote on Reynolds' nomination.

But even if such a move is successful, key senators acknowledged, opponents would filibuster the nomination.

The Republicans are unlikely to bring the matter to the floor unless they can muster the 60 votes needed to break a filibuster -- a prospect made more unlikely by yesterday's Democratic criticism.

The Post article also said that some administration officials, anticipating a filibuster, are considering giving Reynolds an unusual recess appointment that would allow him to serve through 1986 without Senate confirmation. "The president's lawyers know better than that," Byrd said. "That's not what the recess appointment is for." Sources said that if the administration promotes Reynolds in August, it would also have to give recess appointments to several top Justice Department officials because an angry Senate would refuse to approve them.

"You wouldn't even get a U.S. attorney through," a Senate official said.

Despite yesterday's negative reaction and the considerable obstacles in their path, some administration officials remain adamant in their desire to promote Reynolds.

These officials, led by Attorney General Edwin Meese III, have argued strongly in the administration that the fight for the nomination should be continued.

First, they believe the Reynolds' defeat unfairly tarred the president's civil rights record.

Both White House and Justice Department officials are convinced that Reynolds was "nit-picked to death" by opponents who seized on discrepancies in his statements, rather than challenged on what they see as the heart of the administration's civil rights policy, its opposition to racial quotas.

Second, several administration officials contend that the president needs to have the Judiciary Committee "in line" before a possible Supreme Court nomination. They anticipate at least one opening soon, possibly to succeed Justice Lewis F. Powell Jr. who was recently hospitalized. "The tail is not going to wag the dog on these nominations," said a White House official. "That committee has to understand who is the president of the United States and this is not a good experience to have as we approach bigger battles.

"We expect to get our people confirmed, not to expose them to shooting-gallery politics that embarrasses the president. There will not be a second Brad Reynolds . . . there shouldn't be a first."

Many in the administration believe they should not be in a position of defending their policies after a landslide reelection victory last November. They also want to respond to negative votes by Specter and Mathias.

"Winning starts with having your own team in line," a White House official said.

But some leading Republicans said the White House should accept defeat rather than spend more political capital on a fight it is unlikely to win.