Donald Trump’s stature within Republican ranks appears to be waning, after most of the party’s presidential hopefuls have shot down the billionaire developer’s invitation to take part in a “debate” moderated by “The Donald.”

Texas Gov. Rick Perry and Rep. Michele Bachmann, of Minnesota, became the latest to announce they will take a pass on the Dec. 27 Newsmax-sponsored debate, following in the footsteps of Texas Rep. Ron Paul, of Texas, former Utah Gov. Jon Huntsman and former Massachusetts Gov. Mitt Romney. Former House Speaker Newt Gingrich and former Pennsylvania Sen. Rick Santorum, meanwhile, have said they plan to attend the event.

The announcements from the Perry and Bachmann camps coincide with a Rasmussen Reports poll released this week that found voters aren’t overly enthralled with the idea of having Mr. Trump involved in the GOP nomination race.

“A sizable number of voters think Republican presidential candidates are paying too much attention to Trump, and an overwhelming majority say his endorsement would hurt a candidate in their eyes or have no impact,” the survey of of 1,000 likely voters found. In all, 42 percent of the respondents said the GOP White House hopefuls have paid too much attention to the reality television star and New York real estate magnate...