FBI Director William H. Webster's conversations with presidential counselor Edwin Meese III and national security adviser Richard V. Allen concerning the FBI investigation of $1,000 in cash Allen received from a Japanese journalist have raised the possibility of a Justice Department investigation of Webster--a man with one of the cleanest reputations in Washington.

The Justice Department's internal investigators in the Office of Professional Responsibility could open an inquiry into whether Webster's conduct violated any department rules or regulations.

Michael E. Shaheen Jr., head of the office, refused to comment yesterday.

His is the same office that investigated and exonerated former attorney general Benjamin R. Civiletti last year. This followed disclosures that Civiletti had asked President Carter to urge his brother, Billy Carter, to register as a foreign agent for the Libyan government during a Justice Department investigation of Billy Carter's ties to Libya.

Attorney General William French Smith last February released guidelines on outside Justice Department communications concerning pending investigations. Those guidelines were similar to ones developed after the Watergate scandals to protect the department from outside political pressures, particularly from the White House and Congress.

Specifically, the Smith guidelines state that all questions from the White House staff or from Congress concerning pending investigations or cases should go directly either to the attorney general or the deputy attorney general.

The guidelines add that any communications from the White House about specific investigations or cases should come through the office of White House Counsel Fred F. Fielding.

Former attorney general Elliot L. Richardson said yesterday that he issued the first set of guidelines in 1973 in part at least because of evidence that the White House had been involved in "exerting improper political influence on the Justice Department."

Shortly after his appointment came revelations that acting FBI Director L. Patrick Gray III and others in the Justice Department had been secretly turning over confidential Watergate investigative material to presidential counsel John Dean.

Ironically, both Fielding and Allen were White House staff members in the early Watergate days. Fielding, who served as Allen's lawyer before both came to the White House for a second time, was assistant to Dean.

So far the Justice Department has refused to explain or justify Webster's two communications outside of the established channels.

Yesterday Common Cause, the citizens' lobbying group, sent a letter to White House chief of staff James A. Baker III asking the White House to assume responsibility for an inquiry into the Justice Department conduct and into whether Allen violated standards of conduct that would not be covered by the Justice Department investigation.

"We've been through this before," said Common Cause President Fred Wertheimer, who sent the letter to Baker. "The question is whether these kinds of contact can cause serious problems . . . . The answer is yes. They've caused serious problems in the past."

Civiletti, however, said yesterday that it is possible Webster has not violated the guidelines.

He said that because Meese had originally asked the Justice Department to look into the $1,000 found in Allen's safe, "and he had the responsibility whether the information or the allegations were such as to disable a top White House aide, it would have been a duty on the part of the attorney general, the deputy attorney general or Judge Webster to inform Meese whether Allen was disabled or not.

"From what little I know, it does not seem extraordinary or way out of line for Judge Webster to have so advised Mr. Meese--if that was the nature of the advice," Civiletti said.

Webster's other call was directly to Allen, and he reportedly told Allen that other participants in the investigation had backed up his version of the story--that he had received $1,000 intended for Nancy Reagan from a Japanese women's magazine and that he had put it into his safe where he forgot about it.

Civiletti said it would be "hard to speculate on the call to Allen . . . . It's not unusual for the bureau or someone high up in the bureau to call and tell a high-level subject in an investigation that 'the bureau is seeking your cooperation . . . . You can expect further inquiry,' " Civiletti said.

But some Justice Department attorneys have privately expressed dismay that Webster would discuss the case with either Meese or Allen. "Those calls were totally out of the chain of the investigation," one said.

Another prosecutor said he felt Webster might have let his guard down because he thought the investigation was over. But "it was dumb" for him to have made either call, he added.

Richardson said yesterday that he had put the regulations into effect in 1973 because of major "problems involving public confidence in the Justice Department" following the Watergate disclosures and the 1972 ITT scandal in which President Nixon called then-Attorney General Richard G. Kleindeinst instructing him not to appeal a lower court decision dismissing the government's attempt to break up ITT.

He said he issued his order on outside communications to protect the department "from that kind of pressure." He added that he believed the existence of such an order, complete with a requirement that all employes record contacts with outside parties, "would be likely to discourage approaches to the department by persons not confident of the purity of their motives."

Richardson resigned as attorney general when Nixon ordered him to fire Watergate special prosecutor Archibald Cox.