It's not an Internet myth, that's for sure! (Though I appreciate my work being elevated to the potential level of "Internet myth"!) I've been researching them for several years, and he definitely pushed for them in 1954. The sources cited are valid in the article (I wrote it), but there are other documents as well.

Without going into too much detail, the basic timeline is that Teller pushed for 10GT bombs in 1954, and work took place on the 1GT version at Livermore in a serious way (in that reports were generated, etc.), and eventually the USAF reported it did not need or want a 1GT weapon. Eisenhower in general shut down work on "very high yield" weapons (the lower end of which being considered was 60MT). After the 100MT Soviet detonation of 1961, Teller began pushing for a 1GT weapon again, and the labs both pushed for 100MT bombs, but the Kennedy admin basically decided that it couldn't both have 100MT (much less larger) bombs (which it wasn't sure it wanted) and the Limited Test Ban Treaty, and shut down that work as well.

There is more to the story, but it's an interesting trend that these "very high yield" (their term) bombs were always shut down by the Presidents (not the labs), were always proposed by the labs (not the military), and never got beyond the drawing board. But they were on the drawing board. Which is interesting enough!

As for more backstory than that... eventually there will be a book by me on this topic, if all goes as planned! So you'll just have to wait for the full thing! :-) But if you're feeling like there isn't much out there on this — you're right!! I have spent the last 5 years researching it, filing FOIA requests, MDRs, etc., tracking down every scrap. This is the kind of thing I often get back from the labs for my efforts (GNOMON was another high-yield weapon in the GT range) — you can see that there is a lot blacked out! (Every page, other than the title and date, in fact.) Weaving these scraps into a coherent story is indeed the difficult (and fun) part of the work.

(Also, the "malicious link" is not malicious. It is just dead, because the DOE took down the database that it was hosted on. Very frustrating. I have reconstructed the database but it is not quite ready for prime time, much less Reddit. But here's the document though, and I updated the link. Quote is on page 58. I have several copies of the document, differently redacted — one makes it clear the code names are Gnomon and Sundial. Sundial as a codename is declassified in the table of contents. A lot of this work involves tracing codenames, when almost everything is blacked out. In this copy of the report, page 37 of the PDF is where they discuss the weapon — entirely redacted. But as I said, I have several versions. In another, Teller's discussion is revealed to have been prefaced with the following quote: "Turning to another topic, Dr. Teller said he wished to comment on the possibility of much bigger bangs.")

Well, the man was apparently pretty nuke crazy. I've heard him described as the "anti-Oppenheimer." He is considered one of the fathers of the thermonuclear bomb along with Stanislaw Ulam and pushed for the design fairly hard.

I haven't been able to find any hard proof he pushed for a 10GT weapon, but it would have been possible to build one using the Teller-Ulam design, just entirely impractical. The way thermonuclear weapons work, in an extremely simplified explanation, is a fission first stage powers and enhances a fusion second stage. So you are essentially using a nuclear bomb to set off a larger nuclear bomb!

All US weapons are known to use just two stages. The Tsar Bomba, a 50-100MT bomb whose 57MT test was the single largest man made explosion in history, was a three stage bomb (thought it's theorized it was several smaller third stages rather than one large one). However, it was also a 26 foot long, 60,000IB piece of metal (the bomber had to have its bomb bay doors and fuselage fuel tanks removed to fit the bomb). It was massive.

So a 10GT weapon is definitely possible. You just add more stages to the bomb. But the resulting monstrosity would have been far too large to deliver anywhere and would have essentially been a literal doomsday device. The resulting fallout would have potentially destroyed Earth's ability to harbor complex life. It would have no purpose other than revenge after the fact and wouldn't accomplish anything an all out nuclear war wouldn't do on its own.