In the scholarly works on Irfan that I have studied, I have not seen any discussion underlining specific connections between the two terms. The two concepts are also discussed under two different and relatively distant fields of Islamic sciences.
Nafs as human soul is discussed mainly in philosophical works concerning the nature of human consciousness, and is also the central focus in "knowledge of the self" as an important theme in practical ethics (الأخلاق العملي) practical philosophy (الحکمة العملية) and practical mysticism or irfan (العرفان العملية). Whereas Nafas ur-Rahman is a concept in Theoretical Mysticism which discusses the essence, attributes and acts of God.
But due to the possible common lingual roots of the two terms, and also the similarities that exist between the characteristics of the two beings as discussed in respective fields of Islamic knowledge, we may reasonably hypothesize or highlight some implicit connections.
In both scientific and common usage, nafs refers to the incorporeal substance in every living creature that is the source of consciousness, life and power. Nafas is also a vital sign of most living creatures. That is nafas is the sign of nafs. I personally have no knowledge on the etymology of these similar sounding words. But it is very likely that they might have identical roots and therefore even similar meanings on the lingual level.
Nafs similar to Nafas ur-Rahman, is a being that sustains and permeates its subject just as Nafa ur-Rahamn as the First Principle sustains and permeates all consequent levels of creation (not just human soul).
Nafas ur-Rahman as defined in Irfan, therefore, excels and dominates the Divine breath that animates human souls according to the Quran. Because the former is the source of all being, from angels and human souls all the way down to the lowly matter.