Foreign Minister Ara Aivazian of Armenia said in an interview on Wednesday that “the recognition by the United States will be a kind of moral beacon to many countries.” “This is not about Armenia and Turkey,” Mr. Aivazian said. “This is about our obligation to recognize and condemn the past, present and future genocide.” The designation and whether Mr. Biden would issue it have been seen as an early test of his administration’s dealings with the government of President Recep Tayyip Erdogan of Turkey. The two men have had a somewhat testy relationship in the past, in contrast to the generally warm treatment Mr. Erdogan received from President Donald J. Trump, and the genocide declaration could prompt a backlash from Turkey that risks its cooperation in regional military conflicts or diplomatic efforts. Past American presidents have held back from the declaration for that very reason, and Mr. Biden could still change his mind about issuing it.