Dempster, 48, said partner Tracey had turned her life “upside down” and the couple had entered in to a civil partnership in 2007.
She said Tracey, who heads a charity in Glasgow’s Castlemilk district, was now the most important person in her life.
Speaking on BBC Radio Scotland’s Stark Talk, broadcast today/yesterday [WED], she said: “Like many people who eventually realise at a point in their life that they are gay, you go through challenges, and I did struggle a little bit, if I’m honest.
“I probably realised in my mid-twenties. Yeah (I’d had boyfriends) but I met somebody who turned my life upside down entirely unexpectedly.
“It was difficult and it was unexpected for me but I’m really glad it happened because I’m very settled, I’m very happy, and I feel very fortunate. I actually feel pretty blessed.
“It was Tracey. It never happened immediately for us, but we kept in touch and reasonably quickly afterwards we got together and we’ve been together for a long time.
“Absolutely (it’s love). She is the most important person, without question, in my world, and will remain so until the day I die I hope.
“We had a civil partnership in 2007… I won’t lie, it was great.”
Despite working within football, an environment sometimes associated with homophobia, Dempster said she had never encountered it, directly.
She added: “I’ve been lucky in life because I’ve worked in very progressive environments. I worked in media and advertising (previously) and I think because of my toughness I’ve never experienced anything directly.
“I’m certain there was some whispers, some comments, but never really directed at me and I would say if someone were to do that then they would be getting short shrift.”
Dempster joined Hibernian as chief executive in 2014 after holding a similar role at Motherwell FC.
But she revealed that she grew up in the east end of Glasgow in a Rangers supporting family — although her mother was Catholic and her late father a Protestant.
She said: “My family were Rangers supporters and the vast majority of my family still are Rangers supporters.
“I went to my first game when I was 16. My brother had a season ticket and he wasn’t going to a match and I went, and I really enjoyed it.
“My dad was interested in junior football so I would go to junior football matches with him. To be honest, I’ve loved football ever since.”
Dempster agreed to join Hibernian just before the club was relegated to the second tier of Scottish football. She revealed she was offered the chance to walk away, but refused.
She said: “The chairman here at Hibernian, Rod Petrie, after the club was relegated, called me that evening and said ‘this wasn’t what you signed up for, I accept that it’s not the discussion that you and I had’.
“I thought, ‘you’re having a laugh… I’m not leaving now. There’s not a chance I’m going to walk away when things are tough, if anything it makes me more determined.”
As the female chief executive of male-dominated football clubs, Dempster admitted she tends to avoid the dressing room, but admitted players had played pranks on her.
She said: “I tend to try not to go in to the dressing rooms and areas like that, for fairly obvious reasons.
“Sometimes there have been a player or two in the past who have intentionally waited a wee bit longer to get dressed or whatever. I usually send somebody in to make sure the coast is clear.”
Asked by host Edi Stark if players were trying to convert her, she joked: “They may well be. Most of them I’m old enough to be their mothers.”
She added: “As part of the role here, you can be as close to players or as distant as you want, and I choose the former. I’m involved in all the signings, I know them all really well and take an interest in them because ultimately they are a massive part of the club. But they do sometimes play the odd trick on me.”
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