Washington lawyer Abbe D. Lowell said yesterday he has begun his investigation into allegations of misconduct by three top Alexandria city officials in connection with a drug investigation earlier this year.

"There is a cloud hanging over the city," Lowell said in an interview. "I've got to get on with this. There are too many high officials involved for the city to be able to function properly."

Even as Lowell made his announcement, some members of the Alexandria City Council, which ordered the investigation Saturday, began quarreling publicly over how extensive it should be.

The council voted unanimously to hire Lowell to investigate allegations that Director of Public Safety Charles T. Strobel prematurely called off a police drug investigation after the names of two city officials were mentioned in a recording taped by a police undercover agent. In addition to Strobel, the council asked that City Manager Douglas Harman, Strobel's superior, and Alexandria Sheriff Michael E. Norris be investigated.

Lowell was not expected to begin the investigation until after Jan. 8, when the council is expected to approve the terms, scope and payment for his work. The lawyer said yesterday his investigation could take 12 weeks or more and said he began work early in hopes of making certain that all evidence is preserved.

City Council member Donald Casey, a Democrat, said yesterday that Lowell's work must be independent and "no holds barred," a view that contrasted with that of council member Robert Calhoun, a Republican.

Calhoun, a transportation lawyer, said that while he wanted to "clear up these allegations because they have already harmed the city greatly," he was wary of a long investigation. " . . . We do not want him Lowell to go off on some fishing expedition. We just want to know the facts."

That brought a sharp rejoinder from Casey, also a lawyer. "If any members of council want a narrow investigation, there is going to be a big fight," he said. "We are committed to this job. We can't put a dollar figure to it."

Mayor Charles E. Beatley, a Democrat, said he has told Lowell that he would prefer an investigation "confined to the allegations that are now on the table.

"We aren't out to cure all the ills of city hall in one fell swoop," the mayor said. "We just want to clear the air by probing where it's absolutely necessary."

Alexandria Commonwealth's Attorney John Kloch, a Democrat, also voiced reservations yesterday about the extent of the investigation and the role of the City Council in approving it.

"This is an endless cycle of investigations without one solid piece of evidence," Kloch said in an inteview. "The chief of police investigates the sheriff. The city investigates the chief of police. Council wants to investigate the investigator . . . "

The prosecutor said the Lowell investigation "is not an appropriate use of resources" and has been blown out of proportion by the council. The issue really is a question of a "judgment call" by Strobel, who ended the drug investigation. "He Strobel thought there was not enough evidence to continue the investigation," said Kloch.

Strobel has defended his actions, Norris has said he welcomes the investigation and Harman, the city's top administrator for nine years, has declined comment.

The Alexandria Port Packet, a weekly newspaper, triggered the calls for the investigation with an article saying that Sheriff Norris, a Republican, and a high ranking police official were mentioned in the recording taped by an undercover agent. Kloch, who has listened to the tape, has said it does not contain any incriminating evidence against Norris.

The agent was conducting a police drug investigation at Marco's Cafe, a popular Alexandria restaurant, believed by police to have been a center for cocaine sales. A new owner took over the restaurant in February. Some council members say they are disturbed that Harman apparently failed to inform them about the drug investigation. The council members said they needed to know about it because they then were considering merging Norris' department with the police department.