Washington (CNN) Defense Secretary James Mattis has privately told Congress the Trump administration's Pentagon budget request isn't sufficient to cover the cost of rebuilding the military as President Donald Trump has vowed to do, four sources familiar with the conversations told CNN.

Trump has repeatedly said he would rebuild the military with a massive defense spending increase, but the funding planned for next year's budget is less than what the Pentagon sought, according to sources with knowledge of the deliberations.

Mattis is not publicly raising concerns about the $603 billion Pentagon budget plan, aligning himself with the White House's decision, though it's a stance that's sparking frustration from some Republican defense hawks in Congress. But the Pentagon's private assessment matches lawmakers' public criticisms of Trump's budget plan.

President Donald Trump speaks to the press with Secretary of State Rex Tillerson and Defense Secretary James Mattis at the White House on March 13, 2017.

"Mattis continues to express to members of the Armed Services Committees that he's being thwarted getting his message out that $603 billion is insufficient to do what Trump has called for," said a Republican lawmaker, who requested anonymity to speak candidly about internal deliberations.

Trump has said he wants to boost the military by adding tens of thousands more Army soldiers, grow to a 350-ship Navy and add supply the Air Force with more fighter jets.

"Our military is building and is rapidly becoming stronger than ever before. Frankly, we have no choice!" Trump tweeted Sunday. The military, in fact, is still operating under spending levels approved by Congress while President Barack Obama was in office.

In a statement, Pentagon spokesman Lt. Col. Eric Badger said, "We expect Congress will work with the administration to fund much of our additional (fiscal) 2017 budget request. The secretary and the service chiefs highlighted the readiness needs of the armed forces in their recent testimony. That has not changed."

While the White House has touted its $603 billion defense budget as a 10% increase of $54 billion, Republican defense hawks say the White House's math doesn't add up. They argue the defense budget is actually about 3% more than the $584 billion that the Obama administration was already planning for in 2018, and that it falls short of the $640 billion that Republicans like Senate Armed Services Chairman John McCain of Arizona insist is needed for the military.

In closed-door conversations with the defense committees, Mattis has been asked if the funding was sufficient for a rebuilding on that scale. The secretary has told lawmakers a rebuild is not possible, one of the sources said, with the increase only enough to dig out of the military's readiness holes caused by budget constraints over the past several years.

Politically, it's difficult for Mattis to push for more funding in the Trump budget. The Pentagon was the only agency that received an increase in Trump's budget plan, with all other agencies getting significant cuts that have been slammed by Democrats and some Republicans.

Even as Republican lawmakers push for a higher defense budget, the Trump administration's budget plan faces major hurdles -- including lifting the 2011 budget caps for defense, which Democrats have long blocked so long as they don't receive equal increases for domestic spending.

There's also some doubts the military really faces the readiness crisis Pentagon leaders have warned of, with former Obama Pentagon Comptroller Robert Hale saying in February to be skeptical because the military services were putting their "worst foot forward."

Pentagon leaders have warned that budget constraints over the last several years have lowered the military's readiness levels, where units that aren't deploying overseas are unable to train, and planes and ships have to delay maintenance that keeps them operational.

The Navy, for instance, said earlier this year that two-thirds of its F/A-18 fighter jets are unable to fly due to repair delays or needing spare parts.

"We have done this for several years now where we've continued to maintain the operational tempo, but without the backdrop of sufficient readiness funds to keep the forces back here at home at the top of their game," Mattis told a Senate panel last month. "And it's worsening as we go on."

As the Trump administration prepared its budget outline in January and February, Mattis told the President the Pentagon needed funding at or near the levels that McCain and House Armed Services Chairman Mac Thornberry of Texas have called for.

But the Pentagon chief was overruled by another member of Trump's team, budget chief Mick Mulvaney, according to two sources familiar with the deliberations.

Now head of the Office of Management and Budget, Mulvaney was a fiscal hawk in the House who often clashed with his Republican colleagues over his efforts targeting defense spending.

After Mattis made his case to Trump, Mulvaney convinced the President that the budget should not add to the deficit, which was how the administration landed at the $603 billion number.

In a statement, Pentagon spokesman Lt. Col. Eric Badger said, ‎"We expect Congress will work with the administration to fund much of our additional (fiscal) 2017 budget request. The secretary and the service chiefs highlighted the readiness needs of the armed forces in their recent testimony. That has not changed."

An OMB spokesman did not respond to a request for comment.

The $54 billion increase the Trump administration announced for defense was coupled with an equal $54 billion cut to domestic spending, which kept federal spending under the budget caps agreed to as part of a 2011 law to raise the debt ceiling.

"The core of my first budget blueprint is the rebuilding of our nation's military without adding to our federal deficit," Trump said in a letter attached to the budget blueprint released last month.

The $54 billion increase happens to be the same figure called for by both McCain and Arkansas Republican Sen. Tom Cotton. But they were referring to that amount being added to Obama's 2018 budget plan, not the level under the 2011 budget caps.

In public congressional testimony, Mattis has talked about rebuilding the military as a three-year effort, looking beyond the first Trump budget and into 2019.

Mattis' comments suggest he's accepted the administration's budget plan and is now working within it. But Republicans in Congress want him to fight to change the budget before the full Trump proposal is released next month.

"It is right for a combatant commander to say, 'You tell me my funding level and I'll do the best with it I can,'" said a congressional aide. "That's not the appropriate posture for a secretary of defense."

The frustration from some Republicans in Congress is also directed at Acting Deputy Defense Secretary Bob Work -- an Obama holdover who has stayed in his job until his successor is confirmed -- whom Republicans are suspicious of.

Boeing executive Patrick Shanahan was named last month as Trump's pick for deputy defense secretary, but he still has to be confirmed by the Senate.

The Pentagon is also pushing for Congress to approve a $30 billion funding supplemental that would add to the military's current budget. The fate of the supplemental, which also includes funding for Trump's border wall, is still unclear as Congress works to pass a spending agreement before funding runs out on April 28.