California is the only border state that has not announced a deployment of its National Guard to the U.S.-Mexico border.
In March, the Trump administration sued California over its sanctuary city laws that the state says protect undocumented immigrants. Brown responded forcefully, saying the administration was making up "lies."
Most recently, Trump attacked Brown on Twitter for his pardon of five men facing deportation.
A spokesperson for the California National Guard told ABC News on Tuesday that Brown is still considering the president's request.
Former California Gov. Arnold Schwarzenegger took over two weeks to decide on similar requests to send the National Guard to the border in 2006 and 2010.
There are already around 55 California Guardsmen providing support on the state's southern border.
"These personnel perform a variety of missions, including surveillance support at the San Diego Harbor, horizontal engineering in an effort to repair roads, fences and culverts, and criminal analysis with state and federal law enforcement personnel in support of counter-narcotics and counter-narcoterrorism operations," the California National Guard said in an email.
During a radio interview with KTSA on Monday, Texas Gov. Greg Abbott said that, across all border states, "the goal is to have at least 4,000 deployed here in about a month or two," with Texas bringing its total troop deployment to "more than 1,000."
Texas will increase its number of Guardsmen by about 300 per week until they are "fully staffed up," he said. There are currently 250 Texas National Guardsmen on the border right now.
Arizona Gov. Doug Ducey was present to send off 225 Guardsmen from his state on Monday. An additional 113 troops will deploy on Tuesday.
Last Friday, Defense Secretary Jim Mattis signed a memo that authorized the federal government to pay for the potential deployment of up to 4,000 National Guard troops for Trump's border mission through September.
"The National Guard's efforts will include aviation, engineer, surveillance, communications, vehicle maintenance and logistics support," Chief Pentagon spokesperson Dana White said in a statement last week. "These National Guard members will act in support of Border Patrol agents who are performing law enforcement duties."
The Department of Defense has established a Border Security Support Cell led by Kenneth Rapuano, assistant secretary of defense for homeland defense and global security. The 24/7 cell, comprised of Department of Defense officials, is designed to coordinate the department's efforts with the Department of Homeland Security, White said.
Last week, Customs and Border Patrol (CBP) released data showing the number of border apprehensions in March had risen 203 percent compared to March 2017 and risen 37 percent from February 2018.
However, CBP data shows 2017 produced the lowest recorded number of illegal border crossings apprehensions since 1972. A year-to-date comparison between Fiscal Year 2017 and Fiscal Year 2018 showed apprehensions are down nearly 13 percent.
ABC News' Geneva Sands, Justin Fishel and Luis Martinez contributed to this report.