You probably wake up to Howard Stern — but do you want to go to sleep with him, too?

Over the weekend, word broke that Jay Leno is being pushed out to make room for the younger Jimmy Fallon — but sources close to the radio man tell The Post that Stern, in turn, is being “groomed” by NBC as the new Fallon.

Leno’s contract runs out on the “Tonight Show” in 2014, and he’s on the fast track out, despite his great ratings.

Sure, Stern loves the judging gig on “America’s Got Talent,” but even Piers Morgan and Sharon Osbourne could take just so much of extreme cyclists and dog acts. I never figured it was really about that for Stern — it was about creating a Howard that was palatable for TV.

Howard has been trying to show TV suits for years that he’s the whole package — as opposed to the one he talks about on the radio. He’s shown — on “AGT,” one of the most popular family shows on TV — that he can act like a grown-up and play well with others, if the money’s right.

And, according to a source who knows him well, the suits at NBC have gotten the message.

But why, then, would NBC hire a guy — Stern — who is not only just three years younger than Leno but is too old in TV years to even appeal to himself?

Stern’s hot, younger wife (she’s 40, he’s 59), Beth Ostrosky Stern, told The Post’s Sean Daly that, “We were in a hotel and all these little kids recognize him now as the judge from ‘America’s Got Talent.’ Usually it’s their dads. But now it is screaming little kids. People are now seeing him for who he really is.”

So now that he’s acceptable for prime time, is he ready for late night?

“I always say that to him! He is one of the best interviewers out there,” Beth said.

So can she talk him into trying his hand at staying up late? “I am going to!” she said.

Not that it would take much urging since he’s already tried late-night TV several times before — but always with mixed results. In 1987, in an attempt to replace Joan Rivers’ on “The Late Show,” he made a bunch of pilots, but his show never made it to air.

From ’90 to ’92, he briefly had “The Howard Stern Show,” on Saturday nights on WWOR-TV (Ch. 9), and for 11 years, beginning in 1994, E! ran a half-hour truncated version of his radio show.

In 1998, Stern was back on Saturday nights with “The Howard Stern Radio Show” on CBS TV stations, but the content was so risque that hinterland affiliates began deserting the ship in droves. Still, it lasted four years.

But that was then and this is now, when broadcast TV is testing boundaries and is getting riskier and more risque than it ever was a decade ago.

So it makes sense. NBC could keep a tamed-down but still risky Howard on in late-late night until he’s a dirty old man himself, and he’d still have young guys thinking he’s the God of strippers and freaks.

His agent, Don Buchwald, says he won’t comment on clients, and a high placed source at NBC denied the rumor and said the source was incorrect. Stay tuned.

Twitter: @lindastasi