Before it settled on AT&T as the carrier for the iPhone in the United States, Apple shopped the phone to Verizon Wireless and was shot down. It’s thought that Verizon didn’t want to make the concessions (including ceding a lot of control) to Apple which AT&T ended up doing. Of course, the mobile landscape was very different at the time, and now it’s hard to argue that the iPhone hasn’t changed things significantly. So it wouldn’t be a surprise if Verizon wanted another shot at the iPhone. And if a new rumor coming out of Italy is true, it may get that shot.

The report, by ITExaminer.com, claims to have “deep throats” in Apple saying that iPhones for Verizon Wireless will be announced “soon.” It goes on to say the deal is so secret that Apple isn’t talking about it at all and has been trying to hide job postings for EVDO and CDMA engineers — two cellular technologies that Verizon Wireless uses but AT&T does not.

Of course, this is very much a rumor, and Barron’s, which is circulating it, is also calling it a “pretty sketchy report.” It doesn’t help that at the end of the report, the author speculates that Apple could be negotiating “more open deals” because chief executive Steve Jobs is out on medical leave. Even if Apple isn’t currently communicating with Jobs (which seems very unlikely, as he said he would be involved in key decisions while on leave), I find it very hard to believe that Apple’s management would be open to doing anything differently than Jobs would want while he’s out.

Of course, Jobs could want a Verizon deal — he obviously did at one point. And that would make a lot of potential customers in the United States very happy. A few weeks ago, blogger Om Malik wrote that he was giving up his iPhone simply because he couldn’t stand AT&T’s network any longer.

While some are suggesting an iPhone on Verizon could happen as soon as this year, Apple’s deal with AT&T was extended until 2010 last summer. Of course, all the clauses of the deal are not known, and you can bet there are a bunch of them. It makes sense for Apple to eventually spread beyond AT&T if other carriers are willing to let it run the iPhone as it sees fit — which they probably are now given the device’s huge success.