WASHINGTON—The White House is preparing to roll out a long-delayed infrastructure rebuilding plan in January, as President Donald Trump’s advisers bet that voters want a $1 trillion road-and-bridge-building plan—even though it is opposed by some lawmakers.

Mr. Trump’s advisers are putting finishing touches on a plan to direct federal spending of $200 billion or more—funds it would propose to offset with cuts elsewhere in the federal budget—to leverage hundreds of billions more from local governments and private investors...