"Imagine when you are six years old and you arrive at a landscape that looks like no-man's land, with a big fake sky and a huge rollercoaster in the middle of it. Everything seemed so tall, so big, so incredible, almost like another planet. It could have been nothing else but fascinating to me."
Of course, this was the film in which Fellini surrounded his screen alter ego with women of all ages and sizes and inclinations, in a male wish-fulfilment fantasy at once naughty and ironic. She does not mention what her dad was up to on the day of her visit.
Despite both her parents being acting stars - her mother is Catherine Deneuve - Mastroianni did not grow up with the idea of following suit. "When I was young I did not really see myself as an actress, necessarily, but I wanted to be part of that world, because it was very inspiring. At one point I thought I might be a technician. I just loved the atmosphere on film sets. I could watch all these people, working together, for hours."
Eventually, though, and without any great encouragement from her parents, the thespian bug bit. But as if learning a lesson from the paparazzi attentions of her childhood, Mastroianni has developed an altogether different, quieter career than her parents - one which, despite almost 50 films in 20 years, has remained mostly out of the spotlight.
That said, a shrewd taste in directors runs in the family. As well as working with venerable auteurs such as Andre Techine (her debut, My Favourite Season, alongside her mother), Robert Altman (Pret- a-Porter) and Raoul Ruiz (Time Regained), she has developed long-standing collaborations with two of France's most idiosyncratic and talented younger filmmakers, Christophe Honore and Arnaud Desplechin.
It is tempting to feel that none of these directors has really worked their way beneath an innate mystery that Mastroianni has, a personality that seems to float just beyond the camera. Now, at 41, she has teamed up with the inimitable, formidable Claire Denis, whose noirish, very disturbing thriller Bastards takes the Mastroianni mystique and turns it into something new for the actress - a femme fatale.
Speaking to her in a Parisian hotel suite, Mastroianni has her father's doleful, soulful eyes and speaks in confident, stream of consciousness English. Despite being attractive, talented, and the daughter of European film royalty, she is delightfully down to earth, a veritable fountain of self-deprecation.
In the first five minutes she admits to being superstitious, anxious and naive, all by way of explaining why she took so long to suggest teaming up with a director whose films had always "mesmerised" her. When finally she did raise the idea with Denis, the director immediately wrote a role specifically for her.
Bastards involves the return to Paris of old sea dog Marco (played by the craggy Vincent Lindon,) after his brother-in-law has committed suicide. His sister blames a corrupt businessman for her husband's death and demands revenge. Marco moves into the same apartment as his prey, and starts to seduce the man's mistress, Raphaelle, played by Mastroianni. Denis weaves in a sex ring, incest and abuse, to make her film creep under the skin and stay there.
"Claire is a sensitive woman, but she is also a tough cookie," says Mastroianni. "And I think that is what I like about her cinema. She says, 'I'm not pessimistic, I'm lucid.' I understand that. She has a very dark vision of life, but what she describes is not science fiction. In Bastards she talks about money, sex, how power gives you a form of freedom that becomes a trap. And that is all a little real, no?"
The appeal of Raphaelle, she says, was indeed that the Denis wanted her to be "a new version" of a femme fatale. "I have always played the girl who is divorced, dumped, a nice neighbour, in love with a man who does not like women, anything but a woman with nice breasts and high heels," she says with a laugh. "Never. And so I was, 'Wow, this woman has got such an imagination."
When she compares the experience to Sigourney Weaver starring in Alien, it is worth remembering that Mastroianni is a lifelong fan of horror films. What she means is that Raphaelle was uncharted territory for her. "It was scary, because femininity and all that stuff - you have to deliver."
But she is possibly being disingenuous when she says the sex scenes were tough - after all, she has played a leather-clad lesbian vampire, in Mike Figgis's experimental Hotel, shooting sex scenes in complete darkness while Figgis filmed with infra-red cameras. Given half the chance, she can be just as daring as her mother, who essayed a similar role in The Hunger.
Mastroianni rightly sees the difference between her character and the traditional femmes fatales from the 1940s. "It's not evil that turns her on, or money, or betrayal. What makes her fatal is that she wants to protect her kid. She is a maternal femme fatale."
Family is unavoidably a big aspect of her own story. Her parents split when she was just three. She lived with her mother in Paris, spending holidays with her father in Italy, and remembers growing up with "a little bit of resentment" towards the pair of them.
"My mother will always say, 'I was there'. And I say, 'Listen, you need to look at your filmography and stop bull******** me, because between 1972 - when I was born - and 1999 you were making 10 movies a year. So you were not there."
She is enjoying the memory of her family spats. "I'm not trying to play Charles Dickens here, but as a child I really suffered from their absence. I was so envious of my friends who had parents doing normal jobs, I wanted to have parents who were working in a post office. But at the same time it was very exciting, of course, when I had the chance to be on set."
She has two children herself, from former relationships - a son, with the sculptor Pierre Thoretton, and a daughter from her marriage to musician Benjamin Biolay. (Rumour has it she is now dating her Bastards co-star, Lindon).
Her father died in 1996, but Deneuve is as busy as ever. "I think my relationship with my mother is developing with age, and motherhood on my side and being an actress myself. I realise how important it is to have your own bubble. It was nothing personal that they were often away, it is just part of the process."
She has acted with Deneuve several times, with another film in the works. "Acting with her is easy, because she is always been very respectful of my being an actress, not her daughter, and speaks to me in the way she does the other actors.
"The luck that I have is knowing her, and knowing that her reputation for being an iceberg is not true. I know she likes to laugh, and she does gardening and cooking, and picking up the dog muck when she takes him out - she is not living in an ivory tower. Imagine if I didn't know her. With my anxiety, if someone told me I was about to work with Catherine Deneuve I would be terrified."
Bastards opens in cinemas from February 14.