Updated

U.S. Rep. Paul Ryan said Thursday he would love to see Mitt Romney run again for president and teased the GOP's former nominee at one point that the "third time's the charm."

Appearing with Ryan at a public event for the first time since their ticket lost two years ago, Romney offered his own good-humored praise by saying that Ryan "wouldn't be a bad president" himself.

Ryan, R-Wis., has said he will wait until after the midterm elections to decide whether to pursue his own presidential campaign in 2016. Romney has repeatedly denied any plans for another campaign for president. He failed to win the nomination in 2008 and then lost the election to President Barack Obama in 2012.

Romney interviewed Ryan about his new book, "The Way Forward: Renewing the American Idea," on the brink of the fall election season. Republicans are driving for the six-seat gain required to grab the Senate majority. Success would put the GOP in control of Congress and dramatically shape the final two years of Obama's term.

They took turns criticizing Obama's record on domestic issues, including the economy, health care and immigration, with Ryan warning that Obama will "poison the well" on immigration compromise if he takes any unilateral action. Romney said Obama sent a message to Russia when he did not act in Syria and that there has been "an explosion of very bad things in the world" since then.

Romney said Obama's foreign policy is "based on his belief that everyone has the same interests" while he himself believes that some world leaders "want to dominate and some are fundamentally evil."

They also blamed Obama and the Democratic-led Senate for doing too little to reach out to Republicans and to act on important issues.

"If people want to actually see action in this country and dealing with problems from education to health care, immigration to our fiscal needs ... they're going to have to vote for Republican senators and ... a Republican president, as well," Romney said.

Ryan told reporters after the question-and-answer session with Romney that he would "love to see Mitt Romney run for president again."

Long before Romney and Ryan took the stage at the Union League Club of Chicago, Democrats said the pairing is a reminder of failure. In a statement, the Democratic National Committee listed what it called their memorable political gaffes, including Romney being caught on video telling donors that 47 percent of Americans would automatically vote for Obama because they don't pay income taxes and are "dependent upon government."

Romney has been campaigning for GOP candidates across the country, most recently for Rep. Tom Cotton this week in Arkansas, and Ryan has been promoting his book. Romney himself reviewed Ryan's manuscript and offered notes. And Ryan even sought advice from former Romney speechwriters and advisers during the writing.

The conversation Thursday centered on policy issues and didn't touch on the more personal parts of Ryan's book, including the fullest account yet of how Ryan, at 15, found his alcoholic father dead in bed from an apparent heart attack. Ryan told The Associated Press recently that the event shaped him as a politician and family man, and figures heavily into whether he'll seek the presidency in 2016.

Although writing a book often is seen as a prelude to a presidential campaign, Ryan has said he's not sure he's ready to spend more time away from Janesville, the small Wisconsin town where he grew up and where he and his wife, Janna, have raised their three children.