From the Selective Service website:

https://www.sss.gov/register/immigrants/

U.S. immigrants are required by law to register with the Selective Service System.

Immigrant Men Are Required to Register

With very few exceptions, all immigrant males between ages 18 and 25 are required by law to register with the Selective Service System (SSS) within 30 days of arriving in the United States. This includes naturalized citizens, parolees, undocumented immigrants, legal permanent residents, asylum seekers, refugees, and all males with visas more than 30 days expired.
Non-Immigrant Foreigners

Non-immigrant men living in the United States on a valid workers visa are not required to register for as long as they remain on a valid visa up until they turn 26. HB1 Visa's and others. If you have received a letter from us requesting that you register, please send us copies of supporting documentation to show you are exempt.

Exemptions From Service

Some countries have agreements with the U.S. which exempt a non-citizen national who is a citizen of both that country and the U.S. from military service in the U.S. Armed Forces. A non-citizen who requests and is exempt under an agreement or bilateral treaty can never become a U.S. citizen, and may have trouble reentering the U.S. if he leaves.

A non-citizen who served at least a year in the military of a country with which the U.S. is involved in mutual defense activities will be exempt from military service if he is a national of a country that grants reciprocal privileges to citizens of the U.S.

During a draft, any claims for exemptions based on any of the above categories would be granted or denied by a man’s Local Board.

In the Event of a Draft

Most are liable for induction into the U.S. Armed Forces if there is a draft. They would also be eligible for any deferments, postponements, and exemptions available to all other registrants.

However, some non-citizens and dual nationals would be exempt from induction into the military if there is a draft, depending on their country of origin and other factors. Some of these exemptions include:

a non-citizen who has lived in the U.S. for less than one year

a dual national whose other country of nationality has an agreement with the U.S. which specifically provides for an exemption