The officials' disclosure notably illustrates a certain tension within the White House over using military resources to step up security at the border against undocumented immigrants, which was highly prioritized in President Donald Trump's 2016 campaign.
"We have our military now at the border", Trump said as the crowd cheered.
In April, Trump also directed the National Guard to support the Department of Homeland Security and Customs and Border Protection as they worked to secure the southwest border.
Trump warned supporters that multiple caravans of migrants were being organized in Central America to travel to the United States.
Asked about the proposal on Monday, the Pentagon declined comment on internal administration deliberations but added that it did not receive a request from DHS to build facilities to house migrant families.
"We're going to have tents".
The first defense official said that in its response to DHS, the Defense Department said that if Homeland Security officials still want the reserve force of United States troops, they should ask the White House to formally grant the Pentagon the authorities to perform those additional functions.
Meanwhile, Gen Terrence O'Shaughnessy, the head of US Northern Command, which is overseeing the deployment, told reporters last week that there were no plans at the moment for the US military to build lodging for migrants.
The Trump administration last week ordered the deployment of thousands of active-duty troops to support Customs and Border Protection Agents in Texas, Arizona and California in response to the caravans.
The request was turned down because the Department of Defense felt that active duty troops do not have the authority to conduct that type of mission unless they are granted additional authorities by the President.
"While DHS has discussed the need for potential assistance with force protection of CBP personnel, calling this line of support "law enforcement activities" would be factually inaccurate", the DHS official said.
This weekend, US troops continued to work along the border with Mexico, ostensibly to protect the United States from what President Trump has called an illegal immigrant invasion.
President Trump sent the troops to the border as part of Operation Faithful Patriot.
Trump's push to send the military to the border comes ahead of Tuesday's mid-term congressional elections and has triggered sharp reactions, with critics calling it a political stunt that misuses USA military resources.