The Civil Aeronautics Board yesterday gave Pan American World Airways the authority to fly between Houston and three major U.S. cities: Miami, Los Angeles and San Francisco.

The action, which adds significantly to Pan Am's very limited domestic authority, will put the U.S. flag carrier in direct competition on those routes with National Airlines, the carrier it seeks to merge with in order to gain a domestic route system.

Meanwhile, the merger plans of airlines continued to move forward despite the recent doubts expressed by board members about the wave of merger proposals.

Yesterday, Pan Am and National announced they have begun negotiating "substantive terms" of a merger under which Pan Am would offer National shareholders $41 for each share, up $6 from the initial offer it made last week. (Texas International Airlines, which has been seeking a takeover of National, has already acquired at least 18.2 percent of National's stock at an average price per share of about $27.16. Pan Am has at least 4.8% of National stock.) A Pan Am National combination would be second in size only to United Airlines.

Pan Am chairman William T. Seawell and National chairman L.B. Maytag said yesterday that the boards of directors of both companies will meet separately next Tuesday to consider the results of the ongoing discussions and the recommenations of management.

Any merger would be subject to the approval by the boards of directors of both companies, by National's stockholders and by the Civil Aeronautics Board and President Carter.

Also yesterday, Western Airlines and Continental Airlines said they agreed in principle to merge by forming a new corporation that would be the nation's seventh largest airline. They had announced in July that they had begun exploratory talks aimed at combining the two airlines.

Under terms of the tentative agreement one share of Western's common stock would be exchanged for one share of the stock of the new consolidated firm, and one share of the capital stock of Continental for 1.175 shares of the new company. Western's preferred stock would be replaced by an identical issue of preferred stock in the new company.

The next step toward consolidation will be the preparation of a definitive agreement for approval by the directors of each company. Like the proposed Pan Am-National merger, a Continental-Western combination would also be subject to approval by the CAB and the president.

The merger plans are proceeding despite the CAB's stated doubts that approval of mergers that reduce the number of competitors in the industry is in the public interest during the current transitional period from strict regulation to one in which entry to the airline industry is relatively free.

In an order issued this week instituting proceedings on Texas International's proposed takeover of National and on the proposed North Central Airlines-Southern Airways merger, the board served notice that it intended to apply a strict antitrust standard in its consideration of the cases. The board said it would ask: are the anticompetitive effects of a proposed merger outweighed by "significant" benefits that can't be satisfied by a less anticompetitive alternative?