John J. Byrne, who was named to head Geico Corp. during the insurance company's darkest days in 1976, yesterday was named chairman and chief executive of Fireman's Fund Corp., the insurance subsidiary of American Express Co.

At the same time, American Express and Geico said they are discussing two potential business links. One would involve Geico taking a piece of the property-casualty (liability) insurance business of Fireman's; the other would allow Geico to market its automobile insurance policies to American Express cardholders.

American Express and Geico officials said American Express' approach to Geico about taking some of its insurance business led American Express Chairman James D. Robinson III to offer Byrne the leadership of Fireman's Fund.

William B. Snyder, now president of Geico Corp., will add Byrne's title and duties as chairman and chief executive of the area's biggest automobile insurance company. Snyder, 56, also is president and chief executive of Geico's chief subsidiary, Government Employees Insurance Co.

Geico writes $995 million in insurance premiums every year, most of it as automobile insurance. Fireman's Fund, the 10th-biggest property and liability insurance company in the nation, writes about $4 billion in premiums, the bulk of it in property-casualty (liability) insurance.

Sources said that a provision in Byrne's agreement with Fireman's Fund will permit him to be a consultant to Geico, which does not compete with Fireman's Fund.

Like many property-casualty insurance companies, Fireman's Fund fell on hard times a few years ago. American Express had to pour hundreds of millions of dollars of earnings into the insurance operation to bolster the company's depleted reserves.

Property-casualty insurance companies faced severe problems in 1982 and 1983 because of intense competition in premium prices, rising claims costs and a decline in interest rates. These insurance companies had cut premium prices dramatically but were able to cover their costs by investing the premiums in high-yielding assets.

When rates fell dramatically in 1982, many property-casualty companies began to lose money. Premiums have been rising sharply in recent years.

The combination of rising premiums and weakened insurance company reserves has sent many companies such as Fireman's Fund in search of richer companies such as Geico to take a piece of the business.

Most states limit the amount of premiums a company can write to some percentage of its reserves -- generally called surplus. Reserves were hit by losses. Now that premiums are rising -- in some cases quite sharply -- insurance companies are bumping up against those premium-surplus constraints.

Fireman's Fund is discussing with Geico what is called a reinsurance quota-share treaty. Under such an agreement, Geico, which is a cash-rich company, would agree to take a certain portion of the premiums written by Fireman's Fund (less a commission to Fireman's Fund) and be responsible for the same portion of claims resulting from the business.

Snyder said the agreement can be profitable for both Geico and Fireman's Fund, which do not have any business relationship now.

American Express recently announced that it will transfer the life insurance business of Fireman's Fund to the parent company and eventually will sell some or all of its stock in Fireman's Fund to the public. Company spokesmen have said that property-casualty insurance is not a "close strategic fit" with American Express' other operations. Those include travel services, a major brokerage firm and a mutual fund company. American Express also is in a joint cable television venture with Warner Communications Inc., which it wants to sell.

Byrne, who was executive vice president of Travelers Insurance Cos. before being tapped to rescue Geico in 1976, will succeed Sanford Weill III. Weill was head of the big brokerage firm Shearson Loeb Rhoades and became president of American Express when the company acquired Shearson in 1981. Weill was a major architect of the travel company's moves in all types of personal finance.

Weill was put in charge of Fireman's Fund when it began to falter. But Weill, an aggressive, hard-driving executive, apparently realized that Robinson was going to remain chairman of American Express for a long time and decided to leave.