The president of Allegheny Ludlum Steel Corp., the nation's largest specialty steel producer, has sharply rejected a suggestion he become a Republican "eagle"--major campaign donor--and criticized the Reagan administration's failure to stop alleged dumping by foreign competitors.

In a letter to David M. Roderick, chairman of the United States Steel Corp., who made the suggestion about the eagle program, R. P. Simmons wrote:

"For this Republican--me--the issue is not whether I plan to become an Eagle, but rather whether I can continue to support the president and the Republican party. . .

"Until this administration clearly assumes a role of advocacy for efficient U.S. industries . . . I cannot support it."

A spokesman for U.S. Steel's Roderick said he had received Simmons' letter, but had not responded to it. To become an eagle, a GOP supporter must give $10,000 or more to the Republican National Committee.

In his letter, Simmons described himself as "a Republican, philosphically and politically. I support and believe in those concepts so well articulated by the president."

Simmons questioned, however, what he described as President Reagan's " 'doctrinaire' free trade position . . . I had hoped that after the election that the new administration would quickly understand the unfairness of the battle in the trade area."

He argued that Allegheny Ludlum has improved productivity consistently in recent years but is being "systematically destroyed" by foreign companies subsidized by their governments.

In his letter, Simmons said, "I intend to seek, locate and support those of either party who are sincerely interested in a 'trade solution.' "