Presumably, it is because he is one of the two leaders of the suitors and Odysseus wants him out of the way. After Odysseus kills Antinous, Eurymachus tries to make a deal with him. Eurymachus tells him that the suitors will make amends for what they have done.

Why does Odysseus accept Eurymachus offer of restitution?

Why does Odysseus kill Antinous first? ... Why does Odysseus reject Eurymachus' explanation and offer of restitution? He believes they should all pay the price of death. Why does Eurymachus decide to rally the men and fight Odyssues?

How does Odysseus respond to Eurymachus?

He promises to treat them as Telemachus's brothers if they fight by his side against the suitors. When Odysseus returns, Eurymachus has the bow. He feels disgraced that he cannot string it, because he knows that this failure proves his inferiority to Odysseus.