Former Solicitor General Archibald Cox, 61, agreed yesterday to serve as the Justice Department’s special prosecutor in the Watergate scandals.

Attorney General-designate Elliot L. Richardson told a press conference that he would name Cox to the proposed $38,000 a-year post if the Senator approved Richardson’s own nomination.

A Democrat, Cox was a member of the late John Kennedy’s brain trust in the 1960 presidential campaign against then Vice president Nixon. He served as Solicitor General, the third ranking post in the Justice Department, from 1961 to 1965 when he joined the faculty at Harvard Law School.

At a separate news conference in Cambridge, Cox said any implication of President Nixon would be reported.

“This is a take of tremendous importance,” he said. “Somehow, we must restore confidence, honor and integrity in government.”

Besides last June’s break-in at Democratic National Headquarters, the investigation will cover all alleged offenses rising out of the 1972 presidential campaign and any other allegations involving President Nixon, his White House employees or appointees.

Speaking at the Pentagon, Richardson, who is still Secretary of Defense, said he was confident that Cox’s appointment would help counter any public suspicions that the White House might try to influence the investigation.

“There wasn’t going to be any influence from the White House anyway,” Richardson declared.

The Senate Judiciary Committee, which has not been entirely convinced of that, set a hearing for Monday to question both men on how much of a free hand Cox will actually have.

Richardson said the former Solicitor General found Richardson’s proposed guidelines completely acceptable, “word for word.”

Senate Democrats dissatisfied with the charter are still expected to press for a change in one key provision that subordinates the prosecutor’s independence to “the Attorney General’s statutory accountability for all matters, falling with the Department of Justice.”

Richardson indicated that he has no intention of yielding on that point. But he did say he plans one major change in guidelines, suggested by several senators, to exempt the prosecutors from any duty of keeping the Attorney General informed of the progress of the investigation.”

Cox’s selection ended a sometimes frantic search by Richardson that lasted more than two weeks.

His first choice, Federal judge Harold Tyler, Jr. of New York City, turned down the job Monday citing his reluctance to step down from the bench, particularly when the ground rules for the prosecutor’s post had not yet been settled.

Another of the four “finalists” for the job, former Deputy Attorney General Warren Christopher, took himself out of the running Wednesday, saying that he saw no “reasonable probability” of securing “the requisite independence.”

The withdrawals served to reinforce doubts about the independence Richardson said the prosecutor would have. They also raised questions abut Richardson’s own prospects for Senate confirmation.

Richardson offered the job to Cox Wednesday evening in a phone call to the West Coast where Cox was giving a University of California lecture.

An expert primarily in labor and constitutional law, Cox had been considered earlier and passed over. Richardson said, because of a relative lack of experience in trial work and criminal prosecutions.

As a consequence, Richardson said, Cox’s chief deputy, to be named as soon as possible will be “a lawyer with extensive experience in litigation.”

Cox served for three months last year as counsel to a special Massachusetts legislative committee investigating charges of judicial impropriety against two state Superior Court judges, Edward J. DeSaulnier Jr. and Vincent Brogna.

The two judges had been implicated in a 1962 stock-swindle case involving Michael Raymond of New York, who first aired his allegations in 1971 in testimony here before the Senate Permanent Investigations Subcommittee headed by Sen. John L. McClellan (D-Ark).

In January of 1972, the Massachusetts Supreme Judicial Court ordered Judge DeSaulner’s disbarment as a lawyer (he subsequently resigned from the bench) and censured Judge Brogna. The court then referred the case to the legislature, which alone in Massachusetts can initiate the removal of judges.

The legislative committee, on the basis of Cox’s investigation, issued a report on April 11, 1972, that urged no further action against Judge Brogna and found the case against Judge DeSaulnier “moot” because he had resigned. The findings stirred some controversy since the committee held no public hearings and Cox did not interview either judge. He declined to say whether he had interviewed Raymond. Judge Brogna remains on the bench.

Cox agreed to serve as special Watergate prosecutor in a phone call to Richardson around 1:45 p.m. yesterday. After notifying key senators, Richardson, anxious not to prolong the guessing over his selection, announced his choice at a 3:30 p.m. press conference while the FBI was still updating its last background check.

The Attorney General-designate hailed Cox as “one of the finest solicitors general in recent history ... and a lawyer of courage, independence and integrity.”

Of the two other “finalists” that he passed over, Richardson said that one, retired New York appellate court judge David Peck, 71, not a Wall Street lawyer, decided he could not take the job because of “urgent commitments to clients of long standing.” Of the other, Colorado Supreme Court Justice William Erickson, 49, Richardson said only that he had not been offered the post.

A one time student of Cox in labor law at Harvard Law School, Richardson said that “anyone who knows him knows he’ll do it right without regard to school ties or, any other association. We’ve never been close friends.”

Asked whether the nominees were related in any way to Edward Finch Cox, President Nixon’s sons-in-law, Richardson started to say no, did a double take and confessed amid a round of laughter.

“I didn’t ask him, come to think of it.”

The two are not related.

At his press conference in Cambridge, Cox said the investigation might take a year, 18 months or longer. He pointed out that the Teapot Dome inquiry started in the Coolidge Administration took six years.

He said he was satisfied that Richardson’s guidelines would “permit sufficient independence to do the job right. And whatever else I shall be, I shall be independent.”