Twenty-six of the nation's biggest banks--including Bank of America, Chase Manhattan and Continental Illinois--announced yesterday that they will form the first nationwide network that will permit customers of one bank to withdraw money from his or her account using an automatic teller machine at any other participating bank.

When the so-called Plus System Inc. becomes fully operational, a Chase Manhattan customer visiting San Francisco, for example, could withdraw funds from his or her Chase account using an electronic teller machine at Bank of America. Through a nationwide data link, Bank of America computers would ask a Chase computer system to approve the withdrawal.

Bank of America is the nation's largest, while Chase is the third biggest and Continental Illinois, Chicago's largest, is the sixth biggest bank in the country.

Customers can use the Plus system to withdraw money from checking or savings accounts, transfer funds between accounts, get cash advances on credit cards and find out the balances in their checking and savings accounts. Because of federal laws, however, customers cannot make deposits in a state other than the one in which the bank is located.

No Washington-area banks are members of the Plus system yet, according to Dennis Dumler, executive vice president of Plus Systems. Dumler said the organization is talking to banks in the area and plans to add one as a charter member of the organization.

Some banks will be operating on the Plus system within two or three months, according to Edward Kuhl, vice president of Continental Illinois. By the middle of 1983, Kuhl said, 1,500 automatic teller machines should be linked to the system. By 1984, 3,000 financial institutions serving 25 million households should be part of the Plus network, officials said.

As banks increase their use of automatic tellers and as customers become increasingly comfortable dealing with machines, nationwide linkups of the sort announced today should grow, bankers predict. Several regional systems already exist. One will form the backbone of Plus and several are being developed in this area.

In addition to the 26 charter members of Plus, other banks or savings institutions may join the system provided they receive the blessing of the charter member that is in charge of its area.

The Plus system will be based in Denver, where the Colorado National Bank already runs a regional automatic teller sharing network for 270 banks in 12 Rocky Mountain States. Those 270 banks are the initial backbone of Plus. Among them they have about 300 teller machines.

Dumler said all 26 charter members should be part of the network within a year. D. Dale Browning, president of Plus, said at a New York news conference yesterday that by 1987 10,000 machines will be linked to one another through the Plus computers in Denver.

Several big area banks are forming regional automatic teller networks that will allow customers to share automatic teller machines at banks in the District, Maryland and Virginia.

One network will connect First National Bank of Maryland, Suburban Trust Co., First Virginia Banks, National Bank of Washington and Madison National, among others. Another network will link Riggs National, American Security, Virginia National, Perpetual American and Columbia First Federal savings and loan associations, Dominion Bankshares of Roanoke and Equitable Trust Co. and Mercantile Bankshares of Baltimore.

Citibank, the biggest bank in New York and the pioneer in consumer electronic banking, still operates more automatic teller machines than any other organization, but Citibank's machines are not compatible with any other bank's machines.

While most machines are activated by plastic cards on which the information is encoded on a strip of magnetic tape, the Citibank cards are laser-activated.

Citibank Executive Vice President John Reed said recently that he does not believe the average bank customer is much attracted by being able to withdraw from a Citibank account when visiting Los Angeles. However, he said, the bank could convert its cards and teller machines if it decided to join an interstate network.

Illinois law will not permit Continental customers to participate in the network. Kuhl said the giant Chicago bank is trying to get the legislature to change the law.