Suri, 35, and actress wife Udita Goswami welcomed Devi last year in January.
The director apparently decided to get the tattoo as he will be travelling across the world for his next film, "Half Girlfriend", and wants to have his daughter's memories with him.
"Half Girlfriend", starring Arjun Kapoor and Shraddha Kapoor, will be shot in Patna, Delhi, Rajasthan, Cape Town, New York. The love story is an official adaptation of Chetan Bhagat's novel of the same name.