Approximately 250 National Guard troops were deployed to the Texas-Mexico border on Friday.
South Carolina Gov. Henry McMaster, a Republican, provided to ship members of his state's Guard as correctly.
Napier said Ducey told the sheriffs he will seek additional federal grant dollars for area law enforcement needs.
Although Homeland Security Secretary Kirstjen Nielsen has declined to indicate how long the operation might last, Trump said last week that he expected to keep the units "or a large portion of them, until such time as we get the wall".
Brown has repeatedly fought with Trump over immigration policy.
Over the weekend, the Arizona National Guard activated a team of planners who will coordinate the arrival of new troops.
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. The governors of California and New Mexico have not said publicly how they plan to respond to the president's request.
"We had a 200-percent increase in traffic across the border", said Ducey.
New Mexico Gov. Susana Martinez's office said it would be deploying 250 guardsmen, including 80 later this week.
Trump talked about last week he wishes to ship between 2,000 and 4,000 Nationwide Guard members to the border, issuing a proclamation citing "the lawlessness that continues at our southern border".
Despite drops in apprehensions of undocumented immigrants over time, Ducey said March saw a surge of unauthorized border crossings and illegal activity.
"Your support will free up the border agents to serve on the border allowing us to put more boots on the ground", Ducey said.
California has not decided whether to send National Guard members to the border with Mexico to help President Donald Trump's deployment of troops to fight illegal immigration and drug trafficking. 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 has said her state will take part in the operation, but hasn't announced a deployment. "The Guard will provide air support, reconnaissance support, operational support, construction of border infrastructure and logistical support".
The Republican Ducey in a tweet Monday said more of the state's Guard members will be deployed on Tuesday. It's now unclear where along Arizona's 370-mile border the troops will be stationed.
Trump has said he wants to use the military at the border until progress is made on his proposed border wall, which has mostly stalled in Congress.