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:
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, by B. Grabmayr and A. Visser, Bulletin of Symbolic Logic, 2021.
GÖDEL'S THEOREM AND DIRECT SELF-REFERENCE, by S. Kripke, Bulletin of Symbolic Logic, 2021.
