Democrats, perhaps optimistically, hope to get Mr. Tipton to sign on to the broad bill they recently introduced, which includes a path to citizenship for the 11 million undocumented immigrants already in the country. Immigration advocacy groups hope to show him that there could be political consequences if he does not take steps to help overhaul the nation’s immigration system.

According to the most recent census, Mr. Tipton’s district, which covers nearly the entire western half of the state as well as part of its southern border with New Mexico, is 24 percent Hispanic. Combined with the large agricultural businesses on the Western Slope, the demographic forces alone would seem to compel Mr. Tipton to support some form of immigration overhaul. President Obama won the state narrowly in both 2008 and 2012, and Mr. Tipton won re-election last year with 53 percent of the vote.

This month, a coalition of immigration advocates, as well as labor and religious groups, inaugurated the “Cost of Inaction,” a voter education and outreach campaign that targets nine House Republicans, Mr. Tipton included, to push for a vote on an immigration overhaul before the end of the year. Though the Senate passed its own broad immigration bill in June with bipartisan support, immigration advocates have become increasingly frustrated with the House, which has made little progress on legislation of its own and is unlikely to take a vote this year.