The two Bollywood stars are extremely close, according to Mid Day.
A source told the paper: "They are constantly in touch. She has, in fact, taken charge of his fitness regime. Abhay is very lazy about going to the gym. She wants him to build on his muscles.
"They're constantly arguing over his laidback attitude towards physical fitness."
Abhay admitted recently that he thinks muscles are "overrated" by movie-goers.
The 28-year-old actress confirmed that she was "constantly in touch" with her cousin, who is seven years old than her.
Both Esha and Abhay grew up with parents in the acting business. Esha is also half-sister to Bollywood veterans Sunny and Bobby.