The prevailing view among prominent Republicans is that Trump still has the time and ability to make the necessary course corrections, especially given Clinton's vulnerabilities. But they see some acute problems in the way he has conducted himself in recent days.
Republican strategist Brian Walsh, a former spokesman for the National Republican Senatorial Committee, quickly took to social media after Trump's remarks on Muslims yesterday: "I don't care if he's the nominee - Republicans should loudly condemn this racist, nonsensical rhetoric by Trump," Walsh tweeted.
Walsh, who does not support Trump at the moment, said in an interview that Republican leaders should not hesitate to condemn comments that are "the definition of racism". "It's very toxic for other Republican campaigns and for the party as a whole," he said.
The remark on a theoretical Muslim jurist followed his repeated comments arguing that US District Court Judge Gonzalo Curiel, who is presiding over fraud lawsuits against his Trump University education business, should have recused himself because "he's a Mexican". Trump says his desire to build a wall on the border with Mexico was in conflict with the judge's ethnic background; Curiel was born in Indiana to Mexican immigrant parents.
Republican leaders including Speaker Paul Ryan and Senate Majority Leader Mitch McConnell, who both support Trump, have criticised those statements.
Former Speaker Newt Gingrich, another Trump booster, also criticised him. "This is one of the worst mistakes Trump has made, and I think it's inexcusable," said Gingrich. He added: "If a liberal were to attack Justice Clarence Thomas on the grounds that he's black, we would all go crazy."
Trump has struggled with both his rhetoric and policies relating to minorities. At a rally in Redding, California, on Saturday, Trump pointed to a black man in the crowd and exclaimed, "Oh, look at my African-American over here - look at him."
Rick Wilson, a longtime GOP operative who has been one of Trump's biggest critics, penned a column to fellow Republicans this weekend warning that candidates up for election will be yoked to Trump: "You own his politics. You own his policies, even the ones that only last as long as the next contradiction. You own the racial animus that started out as a bug, became a feature and is now the defining characteristic of his campaign. You own every crazy, vile chunk of word vomit that spews from his mouth."
Wilson said in an interview Sunday that Trump's comments about Curiel and a hypothetical Muslim judge are "overtly racist" and contradict the freedoms outlined in the Constitution.
Wilson said that it is "mortifying" that Trump would use a judge's race as an excuse for why Trump University has lost court decisions.
"For everyone who cries wolf on racism - and there are a lot of them on the other side - they are now validated forever," Wilson said.
Republican concerns over Trump's campaign extend beyond his comments to his strategy.
After cementing his role as nominee following the May 3 Indiana primary, Trump has taken long stretches off the trail and, when he is campaigning, has focused his time on blue states which still have primaries to come but which are virtually unwinnable for a Republican in the general election.
California, where Trump held rallies over the past two weeks, was last won by a Republican nominee in 1988 by George HW Bush.
There are also concerns about Trump's continued targeting of fellow Republicans. During an event in New Mexico last month, he criticised the Governor, Susana Martinez - the first Latina governor in the nation - only to tell a local newspaper the following week that he would like her endorsement.
