A California institution is making a bid to become the McDonalds of banking.

First Interstate Bancorp. told a meeting of securities analysts here today that it intends to franchise its name and services.

This is the first time that any bank in the United States has sought to sell its products like hamburgers under a financial golden arch. It marks another milestone along the road toward nationwide banking, another effort on the part of regulated banks to compete with nonbanks such as Shearson/American Express that offer financial services throughout the country.

First Interstate's market is expected to be the West, but several banks here are also said to be interested in this concept for the East.

First Interstate is the nation's largest multistate banking company. Based in Los Angeles, it owns 21 banks in 11 western states. A few weeks ago it announced formation of a national network of shared automated-teller machines. Through franchising, it has the capability of increasing the 600 machines it now has to 4,000, serving 20 million customers.

Franchisees would be hooked to this network by telephone lines and satellite. Customers would be able to withdraw funds electronically at any franchise bank across the country. First Interstate also is prepared to sell other technical services and is actively interested in developing franchised brokerage services as well, company officials said.

Each franchisee would be required to change its name to First Interstate Bank of its particular city. However, the parent company would not own the banks or even have an equity in them. Ownership and management would remain in local hands.

This is expected to be an attraction for small, independent banks, which supposedly would welcome a chance to gain expertise and technology without losing control.

A First Interstate spokesman declined to discuss franchising fees, saying this was the most delicate part of the concept. Apparently, different fees will be levied according to the size of the bank and the services purchased.

First Interstate officials said the company will monitor its franchised operations to maintain quality control. If it believes that service is unsatisfactory, the company reserves the right to terminate the contract. However, it has no control over banks' other operations, and can not demand from them a certain level of return on investments.

The selling power of nationwide franchises from hamburgers to real estate has been amply demonstrated: Franchising was responsible for 32 percent of total 1981 retail sales.

First Interstate representatives said the company believes it already has proved the attraction of a nationally recognized name in banking. During the four months before it changed its name last year from Western Bancorporation, it lost 17,000 accounts. In the four months after a change to First Interstate, it gained 100,000.