President Reagan, moving promptly to fill the soon-vacant job of attorney general, has begun searching for a successor Edwin Meese hopes will carry on the administration's conservative agenda with "integrity."

"We are beginning the search immediately," White House spokesman Marlin Fitzwater said one day after Meese announced he would step down by early next month. "We would like to have a candidate to replace the attorney general before he leaves, but you just can't predict what the timetable will be."Meese, who met privately with the president Wednesday, said in a television interview that whoever replaces him in the coming weeks should be "someone who subscribes as I do to the principles of Ronald Reagan and his administration."

The nation's chief law officer also said his successor should meet three key criteria: first, loyalty to the president; second, "integrity;" and third, a dedication to the law. Meese has been accused of leaning too heavily on his relationship with Reagan and for lacking integrity beyond reproach.

Just as his tenure began in 1985 with an independent prosecutor's probe that saw no criminal charges emerge, it now ends with a second one that produced no indictment - a 14-month inquiry that critics say incapacitated Meese as a leader and contributed to a spate of resignations at the Justice Department.

Meese nonetheless reiterated Wednesday that he feels able to leave because the investigation by independent counsel James McKay "vindicated" him.

McKay announced Thursday he has provided a copy of his report to Meese's lawyers and to Charles Fried, the Justice Department's solicitor general.

Fitzwater said Reagan, confronting a department where now the four top positions will have turned over in a matter of months, began the quest for a new attorney general early Wednesday, meeting first with his chief of staff, Kenneth Duberstein, and his White House counsel, A.B. Culvahouse.

Culvahouse will collect "names and candidates from various sources" in the next few days, Fitzwater said, and Meese, whose backing of failed Supreme Court nominee Douglas Ginsburg led to one of the administration's worst embarassments, will be consulted on the question.

As suggestions and speculation began pouring in from all camps, Meese brushed off suggestions Wednesday that the still-secret 830-page report by McKay may yet result in another ethics probe within the Justice Department.