The government's most senior Republican, for one, says he does not share Trump's complimentary view of the Russian leader.
"Vladimir Putin is an aggressor that does not share our interests," House Speaker Paul Ryan told reporters Thursday. "It certainly appears that he is conducting state-sponsored cyber-attacks on what appears to be our political system. That is not acting in our interests."
Asked whether he'd be comfortable with Trump as president, Ryan demurred, "I'll leave it at that."
The discussion followed a Wednesday night national security forum in which the presidential candidates made back-to-back appearances. During the session, Clinton was repeatedly challenged on her controversial email use at the State Department and her vote as a senator for the 2003 invasion of Iraq. She also flushed out several national security priorities if she is elected, including trying to take out the leader of the Islamic State and vowing to defeat the extremist group without putting U.S. troops on the ground in Iraq or Syria.
Trump did little to counter the criticism that he lacks detailed policy proposals, particularly regarding the Islamic State group. He both insisted he has a private blueprint for defeating the extremist group and that he would demand a plan from military leaders within 30 days of taking office.
But he was also harshly critical of the military, saying America's generals have been "reduced to rubble" under Obama. Asked to square his request for military options with that criticism, Trump said simply: "They'll probably be different generals."
The Republican also renewed his praise for Putin and his disdain for President Barack Obama, saying that the Russian enjoyed an 82 percent approval rating. "The man has very strong control over a country," Trump said. "It's a very different system and I don't happen to like the system, but certainly, in that system, he's been a leader, far more than our president has been a leader."
Speaking to reporters Thursday morning, Clinton suggested she agreed with Democrats who say she is being held to a different standard in the White House race.
"I don't understand the reasons for it," she said. "I find it frustrating, but it's part of the landscape we live in."
With just two months until Election Day, national security has emerged as a centerpiece issue in the White House race. Both candidates believe they have the upper hand, with Clinton contrasting her experience with Trump's unpredictability and the Republican arguing that Americans worried about their safety will be left with more of the same if they elect Obama's former secretary of state.
Clinton's argument that Trump is ill-prepared to be commander in chief has been bolstered by a flood of Republican national security experts who have chosen to back the Democratic nominee instead of their own party's pick.
Some of those Republicans will join Clinton Friday for what she dubbed at "working session" on the threat of terrorism. Those attending the event include Michael Chertoff, who served as Homeland Security secretary under President George W. Bush.
Wednesday's forum served as a preview of sorts for Clinton and Trump's highly-anticipated presidential debates. The candidates will face off for the first time on Sept. 26 at Hofstra University.
By virtue of a coin flip, Clinton took the stage first and quickly found herself responding at length to questions about her years in government. She reiterated that she had made mistakes in relying on a personal email account and private server as secretary of state and in voting for the 2003 invasion of Iraq as a senator. But she defended her support for U.S. military intervention to help oust a dictator in Libya, despite the chaotic aftermath.
"I'm asking to be judged on the totality of my record," said Clinton, who grew visibly irritated at times with the repeated focus on her past actions.
Trump stood by a previous comment that appeared to blame military sexual assaults on men and women serving together, but added he would not seek to remove women from the military. And for the first time, he opened the door to granting legal status to people living in the U.S. illegally who join the military.
"I think that when you serve in the armed forces, that's a very special situation," Trump said. "And I could see myself working that out."
The Republican also repeated an incorrect claim that he was opposed to the war with Iraq before the invasion. That assertion is contradicted by an interview Trump did with Howard Stern in September 2002 in which he was asked whether he supported the invasion. He replied, "Yeah, I guess so."