Texas announced Friday that it would send at least 1,000 National Guard members, and helicopters took the first of them to the border.
New Mexico Gov. Susana Martinez told The Hill her state would help, but did not announce what date troops would be deployed.
Under federal law Mr Trump invoked in his proclamation calling for National Guard troops.
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.
The Republican governors of Arizona and Texas have already deployed troops.
"The Arizona National Guard will deploy 225 members of the Guard today to support border security measures", the state militia said in a statement.
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The Republican Doug Ducey made the comments Monday after he saw off some of the 250 troops that Arizona is sending in response to President Donald Trump's plan to send up to 4,000 troops to the border.
Gov. Doug Ducey addressed part of the force at Papago Park Military Reservation.
"I work on the border every single day and we make arrests every single day", said Wilson, one of the leading voices of the union representing the Border Patrol.
Trump had said on Thursday that the final deployment would range from 2,000 to 4,000 troops, and he would "probably" keep many personnel on the border until his promised border wall is built.
It said "arming will be limited to circumstances that might require self-defense" but did not further define that.
Doug Ducey met with border sheriffs about Monday's deployment of National Guard troops to the US border with Mexico.
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Hull says the troops could help with air support, surveillance and repairs of infrastructure along the border.
Governor Doug Ducey said on Twitter that "additional members" would be deployed on Tuesday, without providing details.
In California, Brown's spokesman, Evan Westrup, said Monday that state officials are still reviewing Trump's troop request.
In recent years, the number of people detained crossing the border has fallen sharply and is now at the lowest since 1971, according to Border Patrol data.
"The people that are suffering the drug overdoses, often times these products are coming over the border and through our state", he said.
Apprehensions are nonetheless correctly beneath their historic traits all through the phrases of former presidents George W. Bush and Barack Obama, every of whom moreover deployed the Guard to the border.
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