The world-renowned designer - who was responsible for the Team GB kits for London 2012 - says she has been getting tested more frequently as she nears the age her mother was when she was diagnosed.
Linda died almost 15 years ago, three years after being told she was suffering from the disease at 53.
Mum-of-four Stella, 41, told Harpers Bazaar magazine in the US: "It's really hard because you think, 'I've got to make sure I don't get cancer for the sake of my kids, and I'm going to make sure my sister doesn't get it for the sake of her kids'."
"I try to be really preventative and get regular check-ups," she added, "I'm not OK with death. But I'm a real believer in trying to live in the moment."
Stella also revealed the meaning behind the name of her new fragrance - LILY - taken from the nickname her father Paul gave her mother, meaning Linda I Love You, and said that her mum's death is with her 'every day'.
"My mum's death is definitely the most difficult thing in the world I've ever had to encounter," she said, "It's with me every day, the thought of how one deals with all of the repercussions. I'm aware that it's part of everything I do, in a good way and a bad way."
"I have huge admiration for my mum. I think that I'm keeping alive some of the things she believed in and elaborating on that."
She said that staunch vegetarian Linda was 'totally unaffected', cutting her own hair, and never wearing make-up, and that was a 'big part' of her own underlying ethos in life.
"My mum used to always say, 'Health is wealth.' And for me, I can't think of anything more important or more true."
Stella has four children with her husband of nine years Alasdhair Willis; sons Miller, seven, Beckett, four, and daughters Bailey Linda, six, and two-year-old Reiley.