Last August, in the midst of interviewing Rush Limbaugh, I had what I thought was a constructive idea.

The public is politically polarized. Limbaugh is the most influential, and often the most abrasive, voice on the right. So I suggested that Limbaugh strike a blow for civility by playing a round of golf with President Obama.

“When the president of the United States invites you, you go,” Limbaugh growled. But he was highly dubious that any such invitation would be forthcoming. “Obama’s base would never let him get away with it,” he predicted.

I wasn’t so sure. I had served on Israeli Prime Minister Menachem Begin’s staff in 1977 when Egyptian President Anwar Sadat made his amazingly improbable trip to Jerusalem.

Sure, an Obama-Limbaugh golf date is a long shot. Rush mocks and criticizes the president and his agenda five days a week, three hours a day, often in highly charged language. On the other side, Obama has denounced Limbaugh as an extremist and publicly called on Republicans to ignore him. At the 2009 White House Correspondents Dinner, the president sat on the dais, laughing, as comedian Wanda Sykes joked that she’d like to see Limbaugh die from kidney failure.

So there is no love lost. But Obama is well known for his belief in the power of reaching out to foes. If he can extend a hand to America-haters like President Hugo Chavez of Venezuela and President Mahmoud Ahmadinejad of Iran, why not a rapprochement with El Rushbo of Palm Beach?

The following day I tried to broach the idea to presidential advisor David Axelrod, whom I know slightly from his newspaper days in Chicago. He didn’t get back to me. It seemed safe to say that the president wasn’t teeing up.

Almost a year has passed, and the political debate has grown even more angry and contentious. Rahm Emanuel, Robert Gibbs and other high-ranking White House spokesmen have repeatedly charged — with more than a little justice — that Limbaugh is the de facto head of the conservative opposition. That, however, is a reason to talk to him. As this administration has argued in other contexts, peace is made with one’s enemies, not friends.

And where better than a golf course? Eighteen holes (on a course emptied especially for the president) would take about four hours. That’s long enough for these two men — who seem alien, even exotic to each other — to get better acquainted.

In the course of an afternoon, Limbaugh would, I predict, recognize that Barack Hussein Obama is not a foreign radical who hates America for its historical sins; and Obama would see that Limbaugh is a Reagan conservative, not the reincarnation of the racist radio preacher of the 1930s, Father Charles Coughlin.

The ground rules would be simple. Choosing the course would be a presidential prerogative. I suggest Obama allow Limbaugh to bring the refreshments and cigars. Neither man is an especially good golfer, but they are both highly competitive, and for the sake of amity it probably would be best to allow unlimited mulligans and dispense with a scorecard (don’t worry; they’d both keep score in their heads anyway).

The Obama-Limbaugh golf summit would be announced to the public but kept private and off the record.

Mutual friends (yes, they have a couple) would join them as part of an icebreaking foursome and then peel off along the way to give them a chance to talk alone.

What would they talk about? Another simple rule:

No politics. Obama and Limbaugh are never going to change each other’s minds on the big issues of the day (although they are closer on a few matters than you might imagine; both are hawks on Afghanistan, both support gay civil unions). Instead of arguing, they could use the time to discuss what they have in common. They are avid sports fans, for example. They each vacation in Hawaii. They have great cars and huge airplanes to compare and contrast. They might even get around to discussing their fathers, exceptional men who loomed large for their sons and whose difficult legacies followed their boys into maturity. Obama is said to be a great listener, and I can attest from personal experience that Rush is too. They won’t run out of conversation.

This scenario ends with no joint communique, just a photo of the two of them together. It wouldn’t announce the start of a beautiful friendship or even a ceasefire. But it could signify to millions of overwrought citizens that Obama and Limbaugh, for all their disputes and differences, can still see each other as fellow Americans capable of sharing a beer and a laugh on a sunny day. And in times like this, that’s a message worth sending.

Zev Chafets is the author of a newly released book, “Rush Limbaugh: An Army of One.”