The existence of a Gödel-numbering that supports a strong fixed point was claimed by Kripke in his famous essay *Outline of a Theory of Truth*, **Journal of Philosophy** vol. 72 pp.690–716 (the only online copy of the paper I could locate is on JSTOR, so those of you with an academic connection can easily access it). 

On p.693, second paragraph, Kripke makes it clear that he has a proof of the existence of strong fixed points, but he writes "The argument must be omitted from this outline".

Thankfully, Albert Visser has provided a detailed exposition of the existence of strong fixed points (in the usual language of arithmetic) in his majestic 2002 paper *Semantics and the Liar Paradox*, **Handbook of Philosophical Logic**, vol. 11 pp.149-240. 

An online copy of Visser's paper is available on Googlebooks; see pp.159-160 for the details of nonstandard Gödel-numbering that supports a strong fixed point. 

Here is the link to Visser's paper on Googlebooks:

http://books.google.com/books?id=wwXfHT5ka_8C&pg=PA149&dq=%22handbook+of+philosophical+logic%22+%22semantics+and+the+liar+paradox%22&hl=en&sa=X&ei=6t2ET5SXHebk0QGiv8nGBw&ved=0CDYQ6AEwAA#v=onepage&q=%22handbook%20of%20philosophical%20logic%22%20%22semantics%20and%20the%20liar%20paradox%22&f=false


  [1]: https://www.cambridge.org/core/journals/review-of-symbolic-logic/article/selfreference-upfront-a-study-of-selfreferential-godel-numberings/8BC653C04A7DDECC59814C64CC6D1B19
  [2]: https://saulkripkecenter.org/wp-content/uploads/2021/12/Godels-Theorem-and-Direct-Self-Reference-final.pdf

**UPDATE** (March 28, 2023). The following recent papers explicitly focus on the topic of direct self-reference:

[SELF-REFERENCE UPFRONT: A STUDY OF SELF-REFERENTIAL GÖDEL NUMBERINGS][1], by B. Grabmayr and A. Visser, Bulletin of Symbolic Logic, 2021.

[GÖDEL'S THEOREM AND DIRECT SELF-REFERENCE][2], by S. Kripke, Bulletin of Symbolic Logic, 2021.