I heard somebody whisper please adore me
Oh, this is exciting. According to Henry Spencer of sci.space.history the latest Apogee ``NASA Mission Reports'' book -- reprinting press kits, postflight interviews, and miscellaneous oddities about Gemini 12 -- includes a fifty-page proposal, written in 1965, detailing how to send a Gemini capsule around the Moon by 1967. This was always a slightly mad plan, though it was interesting enough that Pete Conrad pushed very hard for it. His Gemini 11 mission used its Agena satellite docking target to lift its orbit to a maximum height of about 1400 kilometers -- still the record manned orbital altitude, if you don't count Apollo 13 as reaching a very high Earth orbit -- as close to the lunar Gemini as any grownups would allow.
The 1965 proposal would launch a Titan III-C with a modified upper stage and a docking collar. The manned Gemini would launch a few minutes later (!), rendezvous and dock, and lift up to the Moon. I haven't seen the book, so don't know whether they'd be able to orbit. While Gemini was very cool in many ways, it was a cramped little spaceship, and had a lot of glitches with fuel cells and Reaction Control System thrusters that could make a lunar flight worrisome. It would probably be riskier than would be worth to try ... but boy, that would have been cool. And it would have completely upstaged Zond and Apollo 8 (and Gemini always had to stop short of upstaging Apollo).
Trivia: In Fiscal Year 1958, its last full year before becoming NASA, the National Advisory Council on Aeronautics had under eight thousand employees, and a budget of US$117 million. Source: Managing NASA in the Apollo Era, Arnold S. Levine. NASA SP-4102.
Currently Reading: Profiles of the Future (1984), Arthur C Clarke.