President Obama may be trying to make Donald TrumpDonald John TrumpDems flip Wisconsin state Senate seat Sessions: 'We should be like Canada' in how we take in immigrants GOP rep: 'Sheet metal and garbage' everywhere in Haiti MORE the "face" of the Republican party, but it's not going to work, Sen. John McCainJohn Sidney McCainMcCain rips Trump for attacks on press NSA spying program overcomes key Senate hurdle Meghan McCain says her father regrets opposition to MLK Day 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."