Webster is among about two dozen GOP lawmakers who support an eventual path to citizenship for millions of people who are living in the US illegally. These Republicans are facing the daunting challenge of trying to persuade colleagues to follow them.
Most Republicans oppose this approach on citizenship, and there is little political incentive for them to change their minds. Only 24 of 233 Republicans represent districts where more than one-quarter of their constituents are Hispanic.
Even so, some in the Republican Party argue that its future hinges on whether the House finds a way to embrace an immigration overhaul, which is a crucial issue for the country's fast-growing bloc of Hispanic voters.
Supporters of a path to citizenship point to demographic changes and business backing that have helped sway Webster, who for years opposed immigrant-rights legislation, as potential motives for wavering lawmakers to sign on.
"I think as a country we need to do something," Webster said in an interview, echoing the rhetoric of Florida Sen. Marco Rubio and other prominent Republicans. "Doing nothing is amnesty."
The small but growing band of Republicans is trying to strike a balance between conservative activists who want border security and immigration advocates who want a path to citizenship.
Many come from swing districts with sizable Hispanic populations that could make a difference in next year's elections, tipping the balance of power in the GOP-controlled House. The lawmakers also feel the pressure from business interests that rely on immigrant labor.
At the same time, conservative taxpayer groups who typically fund GOP primary challenges have remained largely silent on immigration. Anti-immigration activists have failed to organize large-scale demonstrations or generate the kind of public backlash that killed Congress' last attempt to remake immigration policy, in 2007.