They'd be the primary of an anticipated 250 Guard members from New Mexico to serve on the border.
New Mexico Gov. Susana Martinez's office talked about that larger than 80 troops would deploy later this week.
The Texas National Guard has also deployed 150 members for the border security mission announced by Trump earlier last week.
There is no end date for the deployment, Abbott said: "We may be in this for the long haul".
Trump's call for sending troops to the border came after he was heavily criticized by his supporters for passing a federal budget that only allocated $1.6 billion for border security.
McMaster plans to make a similar offer to Arizona's Republican governor in the next few days. Arizona officials announced they were sending 225 National Guard members to the border Monday and would deploy another 113 on Tuesday.
Arizona Governor Doug Ducey plans to send 338 troops to the border. "You show me somebody who is for drug cartels or human trafficking or this ammunition that's coming over a wide-open and unprotected border here".
President Donald Trump. REUTERS Kevin Lamarque
Doug Ducey met with border sheriffs about Monday's deployment of National Guard troops to the USA border with Mexico.
El Paso Sector Chief Patrol Agent Aaron Hull says troops are nowhere near deploying yet.
Troops from SC have helped out on the border before, offering night surveillance by plane and helping a decade ago to build part of the existing wall on the border, Livingston said. Hull says the troops could help with air support, surveillance and repairs of infrastructure along the border so that Border Patrol agents have more time to enforce immigration law. California Governor Jerry Brown has not said if the state's Guard members will participate.
Mr Brown's spokesman, Evan Westrup, said California officials still were reviewing Mr Trump's troop request.
Trump has talked about he wishes to utilize the military on the border until progress is made on his proposed border wall, which has principally stalled in Congress. Safety Secretary James Mattis last Friday authorised paying for as a lot as 4,000 Nationwide Guard personnel from the Pentagon value vary by way of the tip of September.
"The people that are suffering the drug overdoses, often times these products are coming over the border and through our state", he said.
The National Guard has previously been deployed to help patrol the southern border, including in 2010 under former president Barack Obama, and from 2006-2008 under George W. Bush.
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