The 34-year-old Notebook star was spotted in Los Angeles last Friday (February 27) with 'Esme' written in capital letters on the fingers of his left hand.
Eva recently talked to American Violet Grey magazine about choosing her daughter's name.
"Ryan and I both love the Esmeralda character from the Victor Hugo novel, The Hunchback of Notre Dame, and we just think it's a beautiful name," she said.
"Amada was my grandmother's name. It means 'beloved' in Spanish."
Ryan has not confirmed whether the 'tattoo' is real or just something the proud father doodled on his hand with a biro - but a rather worrying third option is it may be a 'DIY' tattoo.
Ryan already has two tattoos that he drew himself. One is a barcode on his wrist and the other is a monster's hand dropping a bloody heart on his forearm.
If Ryan's latest inking is a bonafide tattoo then it marks a change in his taste for comedic body art.
In 2007 he told The Guardian: "A tattoo should never be meaningful, because at a certain point you're going to hate it, and it might as well make you laugh."
Perhaps becoming a father has made him more sentimental.
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