This short answer interprets "visible" as you can see it with your eye.
As seen in the vicinity of the Earth, the visual magnitude (brightness) of the Sun and Moon are -27 and -13, respectively. That's a difference of 14 magnitudes, or $100^{14/5} \approx$ 400,000 times different.
Neglecting geometrical effects and unusual reflected "flares" off the flat solar panels, if the ISS illuminated by the Moon were at the limit of visual detectability at magnitude +6, the sunlit ISS would be magnitude -8.
Answers to this Quora question mention that Wikipedia's Apparent magnitude article mentions that Heavens above puts the ISS' maximum apparent brightness at about -6, which is less than a factor of 10 smaller than our simplistic model would require.
The Heavens above creator also has experience with calculating Iridium flares which can max out near -10, and the ISS has way, way bigger (though much less reflective) flat surfaces that make flares possible.
Answer: So it's really unlikely this could be done regularly with the unaided eye, though a good sky camera with a substantial aperture could certainly capture the trail if the Moon were not above the horizon in the location of the camera causing sky brightness issues.
From the question What goes into an Iridium Flare prediction model besides the ephemerides?
below "View of one of the Main Mission Antenna. The hinged base is on the right side." Cropped. From here.

From How are the silicon PV cells constructed in the ISS's solar panels? Are they as flexible as they appear here?
below: "Scott Kelly fixing a cooling pump during a spacewalk." from Gizomodo's Astronaut Scott Kelly on Liquid Salt, a Stinky Station, and Sleeping in Freefall. Image credit: NASA/Kjell Lindgren

From @Tristan’s answer:
I can't find a public datasheet for the solar cells (if I find one, I'll edit the post), but I can do my best to describe what you're seeing, with the help of this image I found online (rehosted from http://pages.erau.edu/~ericksol/courses/sp300/images/iss_cells.jpg).

The inset image on the lower right is a closeup of the back side of a single cell as seen through the blanket substrate.
From the question How are the orientations of the ISS' eight independent solar arrays optimized?
below: Screenshot from the Science at NASA YouTube video ScienceCasts: Space Coffee that I just happened to be watching at the moment.

