Clenton C. Jones, a District budget analyst appointed last week to oversee Mayor Marion Barry's expenses and office accounts, won't be taking the job after all, according to officials, who said Jones owes the city more than $20,000 in unpaid water bills for a Southeast apartment building he owns.

It was also learned yesterday that Brenda Williams, Barry's staff director since January, had submitted her resignation effective next month.

Williams said yesterday that she told the mayor in October that she would be leaving. "Right now, I'm just physically tired" of the job, Williams said.

The problems with Jones' appointment and the imminent departure of Williams come after months of staff upheaval and controversies that aides say have hurt morale and given Barry's administration an image of disarray.

"The timing {for Williams} makes it look bad," said one official. The official, however, added that senior aides were aware that Williams planned to leave, while the problem with the appointment of Jones caught many staff members by surprise.

Williams, a former management consultant and staff aide to Donald Rumsfeld in the Nixon administration, plays an influential role in the mayor's office, making key staff decisions, managing the paper flow to the mayor and controlling access to Barry.

She is credited with improving the management of the office and won praise yesterday from Barry.

Aides said there is no clear successor to Williams, Barry's third staff director in two years. Williams succeeded former chief of staff Carol B. Thompson, now deputy mayor for economic development.

Clifton B. Smith, now a services ombudsman, preceded Thompson.

Aides to the mayor confirmed yesterday that a background check on Jones, a veteran political operative in the District who has worked as a budget analyst since 1981, was not done until after he was appointed to replace George Thomas as controller in the mayor's office.

Jones declined to comment on his status, but the mayor's office said Jones had asked late yesterday that his name be withdrawn at least temporarily.

Few officials interviewed yesterday said it was likely Jones would be considered for the job again.

Officials said that part of Jones' problem stems from inheriting an apartment building that had existing unpaid water bills. Few other details about the bills were available yesterday.

The disclosure about Jones proved embarrassing, according to aides, not only because it came after his appointment, but also because the job of controller in the mayor's office was created to clean up fiscal disarray.

Barry acknowledged during the summer of 1986 that his official records were poorly kept.

He later reimbursed the District for about $4,700 in expenses and appointed Thomas, a veteran city employe and certified public accountant, to keep records of spending.

Thomas, who recently asked to be reassigned to his old agency, the Water and Sewer Utilities Administration, improved the bookkeeping but disliked the politically charged atmosphere of the District Building, according to associates who said he was never comfortable in the high-profile position.

In addition to the uncertainty surrounding the controller and staff director positions, staff members recently raised questions about Barry's hiring in October of Sallie Melendez, a former Oakland public relations official who receives an annual salary of $63,185, but who until two weeks ago was given no specific duties.

A day after being given her first official assignment, Melendez went on leave without pay to go on a previously arranged shopping trip to Hong Kong. She returned to the District yesterday.

Melendez now holds the title of deputy staff director but is not in line to succeed Williams, several aides to the mayor said.

In addition, Stephanie Greene, chief assistant to the mayor's wife Effi, recently quit, saying to friends she was frustrated in her job.

Greene reportedly told friends that the hiring of Melendez was among the reasons she resigned.