"The security of the United States is imperiled by a drastic surge of illegal activity on the southern border", Trump said.
"We anticipate a one-day special session", said Brown, who added that she had already been in conversation with legislative leaders about the move. "That's a big step", Trump told reporters at the White House on Tuesday.
Sandoval's spokeswoman, Mary-Sarah Kinner, said Friday in an email to The Associated Press that the Trump administration has not contacted the Republican governor's office about the idea.
But North Dakota Gov. Doug Burgum said his state would "answer the call", just as it has done in other times of need, like historic floods.
The president is allowed to call on National Guard units under two different laws passed by Congress in 1956.
Oregon Gov. Kate Brown and Montana Gov. Steve Bullock, both Democrats, also said this week that they would say no to requests for troops from their national guards.
The governors of Texas, New Mexico, and Arizona, who are all Republicans, expressed support for deployment of National Guard troops, according to CNN. Another Republican Governor, Brian Sandoval of Nevada, has said that calling up the National Guard was not appropriate under the circumstances, thus suggesting that his state would not respond positively to a request for troops from Washington. "As Commander of Oregon's Guard, I'm deeply troubled by Trump's plan to militarize our border".
Arizona's Republican Gov. Doug Ducey said about 150 Guard members will deploy next week.
Under U.S. Code Title 10, Trump could "federalize" the Guard, ordering federally-funded National Guard troops under the Defense Secretary's control to report for "active duty".
While California Gov. Jerry Brown has not spoken publicly about Trump's plan, California National Guard spokesman Lt. Col. Tom Keegan said any request "will be promptly reviewed to determine how best we can assist our federal partners".
Gov. Asa Hutchinson said this week that he had offered the support of the Arkansas National Guard. Whether it was natural disasters or even border issues, Presidents Bush and even Obama sent National Guard troops to various places without consulting with state governors. Around 30,000 Guard members eventually participated, according to a 2008 National Guard analysis, including more than 1,000 each from Kentucky, North Carolina and SC.
President Donald Trump wants to send as many as 4,000 Guard members to the border.
But details of the deployment are yet to be finalized, including the number of troops, how long they will be deployed and how much it will cost.