The federal government will investigate the proposed merger of The Firestone Tire and Rubber Co. with the Borg-Warner Corp., officials of the Justice Department and the Federal Trade Commission said yesterday.

Meanwhile the FTC charged that Borg-Warner and Robert Bosch Corp., competitors in sales of automotive parts, illegally share two members on their boards of directors.

The agency said in a separate complaint not including the Firestone merger that having two men on both boards violates the Clayton Antitrust Act and FTC Act.

The two men serving on both boards - Hans Merkle and Hans Bacher - were members of Bosch's board. They both joined the Borg-Warner board in Jan., 1977, as part of an agreement under which Bosch acquired about 10 percent of Borg-Warner's stock.

The FTC wants the interlocking positions terminated, an injunction against furhter director interlocks between the companies and a promise that Bosch can't name anyone to serve on the Borg-Warner board.

If the FTC finds ultimately that the law has been violated, it can order any or all of the above actions. Borg-Warner said its actions "are entirely within the letter of the law."

The complaint points out that the two firms compete in the sale of auto parts, like carburetor kits, and certain non-auto parts such as hydraulic valves. If competition is threatened by agreement between the two firms, the FTC alleged, " [that] would constitute a violation of the provisions of the antitrust laws of the U.S."

The proposed Borg-Warner merger with Firestone, which the two firms announced Tuesday, would bring together two of the largest companies in the nation, forming a new firm with assets of about $5 billion - placing it among the top 30 U.S. corporations in size.

Although Firestone is nearly twice the size of Borg-Warner in sales ( $4 billion to $2 billion) Wall Street analyts said the details of the proposed merger make it appear to be more of a takeover bid by Borg-Warner.

The merger would form a new holding company in Chicago - where Borg-Warner is presently headquartered - and would give Firestone stockholders preferred stock or debentures of Borg-Warner valued at about $870 million.

The beleagured Akron-based tiremaker faces an estimated $100 million in extraordinary charges this year because of a just-signed agreement with the government to recall and replace millions of its premium steel belted 500 radial tires, found to be unsafe by the Department of Transportation.