Chesapeake & Potomac Telephone Co. signed an agreement yesterday that is expected to lead to the largest telephone rate increase ever for District of Columbia residents--a $40.3 million increase that could raise the cost of almost all local telephone service by nearly 20 percent a month.

In a highly unusual settlement with the District of Columbia's people's counsel, C & P agreed to accept less than one-third of the $123 million increase it initially sought four months ago.

In turn, the company would be able to forgo the traditional lengthy Public Service Commission hearings on the request and begin charging the new rates within weeks.

Brian Lederer, people's counsel, predicted the commission would approve the agreement. He said the PSC actively encourages this type of agreement and recently approved similar settlements governing taxi fares in the District.

Under terms of the settlement, at least 80 percent of basic monthly residential service would be affected. For D.C. customers with unlimited calls throughout the Washington metropolitan area, rates would jump from $8.18 to $9.77. Depending on the type of service, business rates would rise between $1.74 and $2.54 a month. However, the rates for pay phone calls (now 15 cents), directory assistance and the basic economy service would remain the same.

C & P had asked for sharp increases in these services when it filed its rate request last November.

The telephone company said it hoped it could begin charging the new rates by April 1. But Public Service Commission officials noted yesterday that it may take the commission until mid-April before it makes any decision on the agreement.

The rate increase for District telephone users is only one of several large increases C & P has been seeking in the four area jurisdictions in which it operates. The Maryland Public Service Commission is expected to rule on a 26 percent rate increase request later this month. A 28 percent increase is pending in West Virginia. Last fall, C & P won permission to impose a 10 percent surcharge on Virginia phone bills.

The rate requests are part of a massive nationwide strategy byC & P's parent company, American Telephone & Telegraph Co., to shift a greater share of the nation's telephone bill from long distance to local users..

The agreement represents the first time the D.C. people's counsel has been able to cut short the lengthy rate hearing procedures and reach a settlement with a major utility over proposed rate rises.

Although substantially less than the $123 million the company had asked for, C & P spokesman Webster R. Chamberlin said the company signed the agreement because "we need the rate increase right now. If we had waited for the hearing process and then didn't get what we wanted--and traditionally we don't usually get what we ask for--we would have just lost money," he said.

Lederer said the agreement puts an end to what he saw as "a dire threat to the future of reasonably priced universal telephone service." Lederer claimed that if C & P's original request had gone into effect, District telephone bills could have increased to five times as high as the current charges.

In addition to the rate increase, C & P was proposing a major restructuring of its rates to place a greater burden on the residential customer. Under the original proposal, rates would have increased charges for installation and operator assistance sharply. At the same time, it would have imposed a fee for operator verification, or checking a line to see if it's busy or out of service.

C & P had been seeking a change in the way the cost of a local phone call is calculated to base the rate on the length of the call, the distance it must travel over telephone lines and the time of day it is made.

Currently, a customer with limited service is charged on a flat rate per call--no matter how far the call must travel over the metropolitan area.

With the debate over the $123 million rate increase now over, Lederer says the commission can now focus its attention on the more significant restructuring issues.

The agreement ends only the portion of the rate increase that deals with revenue; hearings on restructuring continue. When ultimately decided, C & P may be given permission to redistribute the latest rate increase to reflect any restructuring allowed by the commission.

For residential users, the $40.3 million rate rise is the first since 1976, when C & P was granted a $7 million increase. Rates for businesses were raised last year, however, when a $10.9 million rate increase was granted for special business services.