The congressional commission that has supervised District of Columbia government financial operations since 1976 has tentatively agreed to disband, possibly as early as April, leaving the deficit-ridden city free to manage its own money as it grapples with an acute budget crisis.

Dissolution of the commission would be a major political landmark for the District, eliminating congressional supervision of basic financial operations of the home rule government. But it would also eliminate the support of an organization that has provided indispensable help to a cash-starved and erratically managed city government.

Since it was created at the behest of Sen. Thomas F. Eagleton (D-Mo.) after Congress learned that the city's books were so chaotic as to be unauditable, the commission has disbursed nearly $30 million to revamp the District's budget procedures, install a computerized accounting system, revise collection and billing procedures in several departments and obtain the first full audit of the city's finances.

The existence of the panel, the Temporary Commission on Financial Oversight of the District of Columbia, has made a mockery of home rule, putting such basic municipal functions as water bills, parking tickets and checkwriting under congressional tutelage.

But the commission's decision to allow the District government to assume full responsibility for its own bookkeeping, cash management, budgeting and billing procedures represents less than a full vote of confidence in the city's ability to run itself.

A long report by the commission's staff, assessing the gains made in billing, accounting and cash-management techniques under the commission's direction, warns that they could be undermined by the District's chronic, acute shortages of trained personnel in financial management and data processing.

The staff report describes as "debilitating" a vacancy rate of at least 30 percent in the city's key positions for data processing and financial management, calling it a "serious situation which, if allowed to continue, will further handicap District efforts to overcome continuing fiscal problems."

Executive director Robert F. Stephens put it even more starkly in an accompanying letter to commission members. If the city does not engage private contractors or hire more qualified personnel, he said, "the District will face further degradation in processing of daily financial transactions leading to incorrect financial reporting (and) insufficient management information on essential city programs."

The commission's most conspicuous accomplishment was the issuance Feb. 1 of the first full audit of the city's books. That audit, which showed a cumulative city deficit of $388 million, was critically important because it is the keystone of the city's long-range financial program.

The city plans to pay off its deficit and finance future construction by issuing bonds, which cannot be done without an annual audit to be reviewed by municipal bond-rating agencies in New York. Stephens warned that personnel inadequacies could "place future audits at risk."

Mayor Marion Barry and other District officials have acknowledged that the city lacks the personnel to sustain the complex financial management programs imposed on it by the commission. Congress granted the commission virtual final authority on the establishment of an accurate and acceptable accounting and money management system for the city.

In an effort to aid recruiting for the key financial management positions, Barry has asked the City Council to empower him to grant exemptions to the law requiring new employes to live in the District, but the council has not acted.

Neither the staff report nor Stephens' letter has been made public. They are scheduled to be discussed at a commission meeting Tuesday. At that meeting, commission sources said, the members will discuss procedures for phasing out their remaining contracts and turning over to the District about $9 million that remains unspent of the total $38 million appropriated for the commission's work.

The commission has eight members: Barry, City Council Chairman Arrington Dixon, and the members of Congress most closely involved with District affairs. They are Eagleton, ranking minority member of the Senate subcommittee on governmental efficiency and the District of Columbia; Sen. Charles McC. Mathias (R-Md.), the subcommittee chairman; Sen. Alphonse D'Amato (R-N.Y.), chairman of the Senate D.C. appropriations subcommittee; Rep. Ronald V. Dellums (D-Calif.), chairman of the House District Committee; Rep. Julian C. Dixon (D-Calif.), chairman of the House D.C. appropriations subcommittee, and that subcommittee's ranking minority member, Rep. Stewart B. McKinney (R-Conn.).

They have refrained from public discussion of the audit results and of the commission staff's assessment of what has been accomplished. Commission sources said, however, that they agreed informally that they had done all they could to help the District stand on its own feet.

"Nobody is saying the task is done," one source said, "but we're in a situation where the District should be taking the lead. It's not useful for us to continue this kind of involvement."

The biggest potential obstacle to disbanding the commission is the opposition by the General Accounting Office. The law creating the commission specified that the GAO must approve whatever new systems were established. The GAO has criticized as "woefully inadequate" a key component of the entire program -- the "documentation" or operating manuals for the sophisticated computerized accounting system into which all transactions must be recorded.