Republicans worried about their party’s standing with Hispanic voters have launched an election-year scramble to put a better face on their party’s immigration problem. Florida Sen. Marco Rubio, the son of Cuban immigrants, is working with senators from other immigrant-heavy states like Jon Kyl of Arizona and Kay Bailey Hutchison of Texas on their own version of the DREAM Act to help undocumented children. Kyl and Hutchison have held several closed-door meetings with a key Democrat to see whether there’s bipartisan support for a compromise plan. Republicans are also exploring changes in visa rules to attract more high-skilled workers and...