Government Employees Insurance Co. moved yesterday to seek substantial stock ownership of three affiliated firms, through offers to purchase shares at prices modestly above current market values.

Chairman John J. Byrne said Geico, an automobile insurance firm, would make tender offers for up to $300,000 shares each of Government Employers Life Insurance Co. (Gelico), Government Employees Financial Corp. (Gefco) and Criterion Insurance Co.

The public offerings will begin late this month or early in February, if cleared by regulatory agencies that include the D.C. Insurance Department.

Geigo first began acquiring interests in the three companies last year, exercising on Nov. 1 an option to buy several classes of shares for a total investment of $30.86 million.

In hearings before D.C. Insurance Superintendent Maximilian Wallach on the stock purchases, Geico said it would seek to acquire significant stock ownership in the affiliates to give diversity to Geico's operations and to strengthen ties among the four firms.

Byrne said last summer - and he repeated it yesterday - his anticipation that "these proposed purchases would increase Geico's operating earnings without additional equity dilution."

Geico's announcement yesterday said the company would purchase shares of its affiliates at the following prices (with yesterday's quotations in over the counter trading): $14 a share for Gelico ($12.125); $12.25 for Gefeo ($9.75) and $21.50 for Criterion ($17.25).

The ownership impact of the purchase offers if fully subscribed by current owners, would be as follows:

Geico's ownership of Gelico, now 24.2 per cent of the outstanding shares, would be increased to 30.9 per cent.

At Criterion, where Geico's current stock amounts to 33.4 per cent of the oustanding shares, ownership would be increased to 54 per cent.

For Geico, of which 20.5 per cent of common shares now are owned by Geico, the equity interest would rise to 48.1 per cent.

Although Geico is making no offering for Gefco preferred stock, the purchases last November included 6,600 shares of the preferred issue, or 1.9 per cent of the stock outstanding.

Gelico, Criterion and Gefco were initially capitalized in 1949, 1961 and 1962, respectively, through offerings of stock to owners of the then-existing Geico or affiliated companies.

Gelico specializes in life insurance and Gefoc is a consumer finance company. Criterion also is engaged with Geico in the business of selling automobile insurance but to customers generally considered greater risks than accepted by the older company.

When established, the three companies were designed to offer a variety of financial services to Geico policy-holders and customers.

Although there is a diversity of directors on the four companies' boards, the firms have a majority of directors in common. Each has an independent chief executive but the companies have shared common facilities and cooperated on variois business operations and direct mailings of promotional material - historically the chief source of new business.

Tonight, at a New York supper club, Geico will introduce to the press jazz musician Benny Goodman and the entertainer's new broadcast commercials on behalf of the national auto insurance firm, which has its operating headquarters in Chevy Chase.

Geico is preparing a marketing campaign to boost policyholders for its auto insurance and also for other insurance policies. Byrne has established a goal of increasing non-autobusiness to protect the firm from gyrations in profits in the property and casualty business.

Now completing a year of solid recovery, Geico previously limped through several years of heavy losses and was close to bankruptcy, partially because previous officers had underestimated soaring claims costs for auto coverage.