The country's Senate passed a resolution last week saying Mexico should suspend cooperation with the USA on illegal immigration and drug trafficking in retaliation for Trump's move.
Ranchers living near the Arizona-Mexico border hope National Guard troops will help increase border security.
If 4,000 troops were deployed, that would be about double the current USA military presence in Syria and about half as many as the number of U.S. troops in Iraq.
The troops have been barred from interacting with migrants or any people that Border Patrol agents detain.
According to ABC News, citing an Arizona National Guard statement, the additional 113 troops, which brings the total number deployed to 338, will support federal and local law enforcement agencies to stop illicit trafficking in drugs, humans, and arms, among other crimes.
"Your support will free up the border agents to serve on the border allowing us to put more boots on the ground", Ducey said.
Texas is also sending National Guard members to the southwestern US border.
The original announcement that Arizona troops would be deployed came after Texas announced on Friday it would send 250 National Guard troops to its border with Mexico. Jerry Brown and has not made a public commitment to sending troops from his state's National Guard.
The Texas National Guard had said Friday that 250 of its members would be deployed to the border within three days as part of President Donald Trump's plan to fight illegal immigration with up to 4,000 National Guard members. Some Guard members started arriving at the border as states and the federal government officials continued to discuss what they will do.
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Rios said the added troops are unnecessary and that arrests at the U.S. -Mexico border are at an all-time low. It's not clear when they will head to the border.
The New Mexico Guard members could help with air support, surveillance and infrastructure repairs, Hull said.
He not only called the President's plan a declaration of war but said it sends the wrong message to the immigrant community. Evan Westrup, a spokesman for Brown, suggested the Related Press Monday that state officers are nonetheless reviewing Trump's troop request.
New Mexico's Republican governor says her state will participate as well.
Gov. Doug Ducey suggested a gaggle of troopers making able to deploy from a Phoenix military base Monday that their "mission is about providing manpower and belongings" to assist companies on the border and denied that there was a political motive.
While Guardmembers will be armed, they will not perform law enforcement duties.
South Carolina's governor is offering to send National Guard troops to Texas help fight illegal immigration and drug trafficking along the Mexican border.
South Carolina Gov. Henry McMaster, a Republican, offered to send members of his state's Guard as well. It's now unclear where along Arizona's 370-mile border the troops will be stationed.
Trump said last week that he would be sending up to 4,000 troops to the southern border until his proposed border wall was built.