One official said Trump's request for the Pentagon included more money for shipbuilding, military aircraft and establishing "a more robust presence in key international waterways and chokepoints" such as the Strait of Hormuz and South China Sea.
A second official said the State Department's budget could be cut by as much as 30 percent, which would force a major restructuring of the department and elimination of programs.
The officials requested anonymity because the draft budget had not been made public yet, according to Reuters.
Trump, in a speech to conservative activists on Friday, promised "one of the greatest military buildups in American history."
Some defense experts have questioned the need for a large increase in US military spending, which already stands at roughly $600 billion annually. By contrast, the United States spends about $50 billion annually on the State Department and foreign assistance.
The amounts that Trump is proposing to add to the Pentagon budget and trim elsewhere are not yet publicly known.
John Czwartacki, a spokesman for the White House's Office of Management and Budget, said the budget blueprint would be released in mid-March.
"It would be premature for us to comment - or anyone to report - on the specifics of this internal discussion before its publication," he said in a statement.
The budget plans that the White House is expected to send to departments and agencies on Monday are just one stage in a lengthy process.
The agencies can argue for more funding, and final spending plans must be approved by the US Congress.
Trump's budget assumes annual economic growth of 2.4 percent, the second official said. While campaigning for the presidency last year, Trump called for a "national goal" of 4 percent economic growth.
Treasury Secretary Steven Mnuchin, speaking on Fox News earlier on Sunday, said Trump's budget would not seek cuts in federal social programs such as Social Security and Medicare.