President Obama may be trying to make Donald TrumpDonald John TrumpIvanka Trump pens op-ed on kindergartners learning tech Bharara, Yates tamp down expectations Mueller will bring criminal charges Overnight Cybersecurity: Equifax security employee left after breach | Lawmakers float bill to reform warrantless surveillance | Intel leaders keeping collusion probe open MORE the "face" of the Republican party, but it's not going to work, Sen. John McCainJohn Sidney McCainRubio asks Army to kick out West Point grad with pro-communist posts The VA's woes cannot be pinned on any singular administration Overnight Defense: Mattis offers support for Iran deal | McCain blocks nominees over Afghanistan strategy | Trump, Tillerson spilt raises new questions about N. Korea policy MORE (R-Ariz) said.

"I think he may try to, but I don't think that's going to happen," McCain said on CBS's "Face the Nation" on Sunday when asked whether Obama has been trying to paint Trump as the example of the Republican Party.

Trump has been one of the loudest cheerleaders of the so-called "birther" movement, which contends that Obama was not born in the U.S. and therefore is ineligible to be president. Last week Obama released his long-form birth certificate to try to put the issue to rest once and for all.

Immediately after the White House release, Trump took credit, saying he had "accomplished something that no one else has been able to accomplish."

Democrats are hoping that the more attention Trump's "birtherism" gets, the more Americans will think all Republican candidates indulge in such fringe rumors.

At the White House Correspondents' Association Dinner on Saturday night, a fair amount of jokes by both Obama and comedian Seth Myers were directed at Trump. 

Now that the birther issue had been settled, Trump could focus on other highly unlikely theories such as "did we fake the moon landing? What really happened in Roswell? And where are Biggie and Tupac?," Obama joked.

"I think Mr. Trump is having a lot of fun and it's pretty clear he enjoys the limelight," McCain said. "We have very serious candidates. And I think that, if Mr. Trump wants to run, he's welcome to run."