A close as I can find is Gotham Rounded (Medium)

Odd name, but this typeface has the same general feel (text, not numerals):

http://www.myfonts.com/fonts/abstrkt/fun-city-level/

Reminds me of a very precise architect's printing. No luck matching the numbers yet....

Register Sans Wide Bold isn't bad for approximating everything (with a little custom work on the 6 an 9), save for the 3:

http://www.myfonts.com/fonts/btn/register-sans-btn/

Kevin, I think Register Sans is too Narrow but the numbers on the clock I agree are most likely custom. I am still sticking with Gotham Rounded at this point.

Register Sans Wide Bold (the very last weight on the page) was the suggestion....

But Gotham Rounded is nothing to sneeze at either....

Bob, I like your suggestion of Gotham Rounded. Here it is at book weight:
The numbers are pretty close and would take only a little editing to become a passable substitute. The letters are fairly close as is.
Don

For the digits, you may want to try Adonis Old Style SG by Willard T. Sniffin [1930] available from Spiece Graphics.
/6 and /9 look fine, but /1, /3 and mainly /2 require some editing to match your sample.

Sorry Jean-Louis but I'm pretty convinced this one should be custom. Officine Panerai was an official supplier to the Regia Marina, the Royal Italian Navy, supplying watches and precision instruments. In the first decades of the 20th century Officine Panerai began experimenting intensively with luminous materials in an effort to make instrument dials, sights, and telescopic devices that could be employed reliably in the dark. The main specificity of their first model Radiomir is a 2 layer dial. Bottom disk is plainly coated with a zinc sulphide and radium bromide luminescent combination. Top one is a completely black stencil, cut out with the digits thus needing a specific stencil design.

Thank you all! Custom, that's nothing to be sorry about Ryuk. It's proof that the better things don't come off-the-shelf.

I absolutely love their design and brand ID. It's understated (except the size), but there's a warmth in the colours, in the shape of the letters, etc.

Though it's all a bit more marketing than actual watchmaking genius, it works. Bill Clinton used to wear one back in the days, and maybe still does, and it seems the choice for young investment bankers. Since the case diameters are so large, the watches are highly visible, which makes them first choice for the nouveaux riches who like to show off. Today, even my barber uses one!

Definitely not for me, except that beautiful table clock. But then, who needs tables clocks?

I own one and believe me, it's a beauty!
For the rest, this kind of type always reminds me of Engravers Gothic/Blair/Sackers Gothic.
Some alternatives now: Trio Grotesk, Hermes, Rescue, Arquitecta (not rounded but well balanced), Como (very nice as well but /A needing to be completely rebuilt mirrored /V + low crossbar; very promising digits as well), Affect, Lawyer Gothic, Biondi Sans, Zeppelin, Luxury Gold, http://Capline, Doublewide, Adrianna, Biondi Sans, Halogen, Idlewid, Burin Sans, Sweet Sans (funny to see using a watch dial to illustrate the typeface)
Previous discussion on some related topic.

Ryuk, I believe you, they look great! A bit oversized in my opinion, but you can always wear them on top of a shirt, like Gianni Agnelli used to do. My choice is a PP Nautilus, you can wear it with a wet suit or a tux, and more ergonomic on the wrist. What more does a man need? :)

Thanks for all suggestions, including the aircraft clock post!

It also reminded me of Engravers and the likes. I've been trying to use them, but with certain letter combinations they look very rigid and "dated". As is the case with my sketch on the right. Some of your suggestions might work better.

Thanks Ryuk for the explanation of Officine Panerai dial technology. My only remaining question is about what year did they introduce the design for these particular numbers.

Many interesting font suggestions, but I now believe that none of them, including the Gotham Rounded numbers -- which are too narrow and have mid height waist lines -- capture the flavor of the numbers in the Officine Panerai dial. Rather than picking and choosing number sets that are poor approximations, its probably easier just to make the numbers from scratch. For example:

The numbers not shown on the dial, i.e. _0_, _4_, _5_, and _7_, are guesses based on the known numbers.
Don

My guess on _4_ was incorrect
Don

But there is also a closed _4_

So which is the right one?
Don

Officine Paneristi 3646 ca1940

Last one, showing a dial design based on the Roman Arabic hybrid numerals used in the late 1930s and called Rolex or California. Here we also have the open _4_ The vintage version is the same but not as flashy.
don