WASHINGTON—The Trump administration is planning to send several thousand additional troops to the Middle East and is moving toward a decision to rush billions of dollars in weapons to allies in the region, according to officials and people familiar with the plans.

Top Pentagon leaders and White House officials meeting Thursday agreed to initially send up to 3,000 additional troops, with discussions under way for more to support submarines, planes, drones and anti-missile batteries.

U.S. military officials overseeing operations in the Middle East had requested up to 13,000 additional troops, officials said. The 2,000 to 3,000 troops officials agreed to send are expected to be the first of what may be a larger deployment, the officials said.

The U.S. currently has more than 20,000 military personnel in Iraq, Afghanistan and Syria, along with thousands more that are part of a recent deployment of an aircraft carrier and other ships and bombers.

Acting Defense Secretary Pat Shanahan and Gen. Joe Dunford, chairman of the Joint Chiefs of Staff, met with White House officials. Earlier, Mr. Shanahan said military officials were seeking additional troops, but wouldn’t say how many or for what reason.