President Donald Trump said construction of a wall on the U.S.-Mexican border would start in months, raising questions about who would actually build it.

Shares in some construction, infrastructure and heavy-equipment companies rose on Wednesday as investors weighed which firms stood to benefit from Mr. Trump’s proposal to build a wall along the approximately 2,000-mile border between the two countries, which would count among the country’s biggest infrastructure projects if it comes to fruition.

There are a number of hurdles. Currently 654 miles of the country’s border with Mexico has fencing, according to a U.S. Customs and Border Protection spokesman. Portions of the border pose construction challenges. Much of it is open desert, and in many places, roads would first need to be built to reach the border, and then a barrier constructed.

It wasn’t immediately clear how the federal government would go about seeking proposals from companies, how long a procurement might take and how it would ultimately be funded.

The White House offered no cost estimate for the project, but outside experts say an end-to-end fence could easily cost $10 billion and possibly much more, depending on details. A wall would be even more.