Mexico agreed to revisit U.S. demands for a radical overhaul of the immigration system if its proposed measures to curb migration don’t work, putting it under intense pressure to stem the tide of Central Americans arriving at the U.S. border.

As part of the deal to avert tariffs, President Trump wrote on Twitter on Monday that the two countries had “fully signed and documented another very important part of the Immigration and Security deal with Mexico” that would require a vote by Mexican lawmakers.

The...