Talk about your what-ifs. New York Magazine’s Gabriel Sherman reports this morning that Michael Bloomberg attempted to engage Arthur Sulzberger in talks to buy the New York Times as his final term as mayor in the Big Apple came to an end. Sulzberger shrugged him off, but Sherman says Bloomberg still wants the newspaper in his empire:

Near the end of Bloomberg’s time as mayor, he told Times chairman and publisher Arthur Sulzberger Jr. that he was interested in buying the Times, according to a source with direct knowledge of the conversation. Sulzberger replied that the paper was not for sale.

Bloomberg’s overture, previously unreported, might be one reason why talk of a Bloomberg-Times eventuality has flared up among insiders in the wake of the most recent round of Times’ layoffs. Given the fact that both sides vehemently deny that there have been recent conversations (Sulzberger “can’t remember the last time he spoke with Bloomberg,” said a spokesperson), this may very well be wishful — or apprehensive — thinking being played out in the echo chamber of media gossip.

But it does seem that Bloomberg is in fact interested in the Times and that his interest has not waned. “Mike has muttered a lot about the Times to a lot of people,” a Bloomberg adviser told me.

The strongest advocate for such a deal is said to be Bloomberg’s longtime political adviser Kevin Sheekey. In pushing Bloomberg to buy the Times, Sheekey is reprising a role he played in 2008, when he encouraged Bloomberg to run for president. According to a particularly knowledgeable source, in the summer of 2013, Sheekey held a private meeting with a person close to Sulzberger and asked how a deal for the Times could get done. “Sheekey basically said, ‘we want to buy the company,'” a person familiar with the conversation said. A Bloomberg source told me: “Kevin has long been a vocal enthusiast to Mike about this. Sulzberger needs to go somewhere for money. We know they have a real issue, and Mike is an obvious person he’d turn to.”

Yes, but he’s not the only obvious person. Sulzberger turned to Carlos Slim as a white knight in 2009, lending Sulzberger a quarter of a million dollars to rescue the paper from a sea of red ink. In exchange, he received “warrants” that allowed him to double his stake to 17% of the paper’s common stock last week. The Times wants to use the funds Slim paid for the shares to buy back stock from other stakeholders, although the Times reported losses the third quarter last year and could use the cash. The WSJ noted that Slim wants to play no part in the management of the paper, which is obvious, since the Sulzbergers control almost 84% of the voting stock and control two-thirds of the board. Thanks to the early payoff of the loan and some other dividend payments, Slim is in an even stronger position than he was in 2009 to buy the Times, if he so desired — although, given the performance of large print media companies over the last decade, it’s difficult to see why he would.

What would happen if Bloomberg bought the Times? Before shrugging off the question entirely, consider how the multibillionaire has operated his current media empire. Bloomberg the media corporation does a fairly decent job of straightforward reporting, especially on economic issues, and has a pretty wide spectrum of voices in its reporting. They recently lured Eli Lake and Josh Rogin from the Daily Beast to do their foreign-policy and national-security reporting under the Bloomberg banner, something that would be unthinkable in the Sulzberger organization. Bloomberg the activist would no doubt use the paper as a platform for his nanny-state policies, but that would be no worse than we get from the Times now.

At the very least, Bloomberg might find a way to turn the paper profitable. And the effort needed to make that happen could keep Bloomberg off of the campaign trail, and force him to spend his money in efforts other than gun-grabbing and soft-drink commandeering. The only thing more fun than that kind of takeover would be if Rupert Murdoch convinced Sulzberger to sell out.