There are several ancient coins that match the TS.
Justin Robinson notes theses similarities in his blog, Byzantine Coins, the Shroud of Turin and the Holy Grail.
At some time during the short but distinguished reign of Byzantine Emperor John I Tzimiskes (AD 969- 976), an artist working at the Constantinople Mint was entrusted with the task of engraving an image of Jesus Christ for a new bronze follis. Earlier emperors had depicted Christ on gold and silver coins, but this was the first time that his likeness would appear on a mass-produced circulating coin.
The Emperor’s decision to depict Christ on his coinage instead of his own portrait may have been prompted by an exciting new acquisition. Constantinople had recently taken ownership of the holiest relic in Christendom, a mysterious image of Christ ‘not made by human hands’ but miraculously transferred onto a cloth, it was said, by Christ himself. Although it was considered too holy to go on public display at the time, our coin engraver would almost certainly have been granted the privilege of entering the Pharos chapel of Constantinople’s Imperial Palace for a special viewing in order to capture a good likeness.
The cloth had arrived in Constantinople amidst much rejoicing on 15th August 944 after being acquired from the city of Edessa (today, Urfa in Southern Turkey). According to local legend, it had been presented to King Abgar of Edessa by Jesus’ disciples when he became the first ruler to convert to Christianity. However, when the King died, the city reverted to paganism, and the cloth was hidden to protect it. Workers repairing the city walls in AD 525 stumbled upon it in a niche high above one of the main gates.
The first coins with Christ on it was in the late 7th century.
The first coins to depict Christ were struck almost three centuries earlier during the reign of Emperor Justinian II (AD 692–695). On that occasion, the coin engravers may have made the 800 mile trip to Edessa to see the Mandylion for themselves. Both the gold solidus and the smaller gold tremissis (one third the weight of the solidus) incorporate many intricate details present in the mysterious image. However, political instability in the region may have restricted future access to the cloth, and later designs appear to have been copies of the first strikes. During the Eighth Century, a fierce debate raged through the Eastern Church about whether it was heretical to make images of the Son of God. Many paintings of Christ were destroyed, and no coins were struck bearing his image for over a Century until the debate was resolved.
Coins are small so there is limited room to make an entirely accurate depiction of the image on the TS. But to make such a similar rendition on a 1 cm coin is impressive.
When flipped and viewed alongside an image of the face on the Shroud, the similarities are extraordinary, especially when you consider that our engraver was working on an area little more than a centimetre in diameter.
Hugh Farey, in his blog Follis Follies gives an annotated comparison.
Byzantine Follis (AD 969-976)
“Most striking of all is the distinctive cross shape incorporating the eyebrows, forehead and nose. There is a long horizontal band above the eyes, bisected by a long vertical line that starts at the hairline and extends downwards to become a long nose.  The base of the nose connects to a smaller horizontal line that forms the moustache, which slopes down slightly on the left-hand side.  There is a distinctive mark on the right cheek , and beneath the moustache is a small square and a forked beard.  The long hair, which hangs down on both sides of the face, has two parallel strands of hair at the bottom left of the image.  These features can be seen clearly on the Shroud image, and the result is a coin that resembles the Shroud image far too closely to be dismissed as a coincidence.”
Byzantine Follis (AD 1028-1041)
“Intriguingly, there is a tiny mark in the centre parting of the hair in the forehead that resembles the inverted “3” shaped bloodstain that appears on the Shroud in the same area.  In addition, the coin artist has replicated the way that the long hair appears to bunch at the shoulders.  The eyebrows are represented with a long horizontal line, and there is the suggestion that the right eyebrow is slightly higher than the left.  There is also a wound-like mark on the right cheek , a moustache that appears to slope down to the left  and, most striking of all, a horizontal band across the throat.”
Farey admits there are similarities between the coins and the TS image, but claims it is not authoritative.
Although all these folles show some general similarities, they none of them give us an authoritative version of the epitome they were generally copying.
The question is not if the similarities are authoritative, but what is the most reasonable explanation of the similarities? As is the case with the similarities with the art depictions of Jesus, the coin depictions come from a single archetype. The most parsimonious explanation of artwork and coins is from the TS.
Guilio Fanti wrote a book Byzantine Coins Influenced by the Shroud of Christ . Unfortunately, the price is above my budget.
Justin Robinson discusses coins in these videos: