The Labour leader has been a vegetarian for almost 50 years after working on a pig farm when he was in his early 20s.
Speaking to staff at an outlet of cosmetics retailer Lush, the 68-year-old said: "I eat more and more vegan food and have more and more vegan friends.
"There are quite a lot of vegan MPs - not a lot but there are some.
"One of my close relatives has just become a vegan and I went to her house for dinner and it was absolutely brilliant.
"So I'm going through the process alright? I won't go any further than that."
But like many vegetarians, his soft spot for cheese could be a potential deal-breaker.
Last year he said: "The variety of cheese in England is amazing. I like Somerset Brie, it has a nice taste to it. And Shropshire Blue, the county where I'm from. The sharper the better."
It's not just fear of missing our your favourite foods that can be an obstacle. Some who have tried veganism have found it difficult when they go out to eat.
Twenty-five-year-old Sophie Howard lives in South Yorkshire and told Newsbeat it wasn't worth the hassle.
"It just became a pain. Sometimes if I wasn't able to find a vegan option I'd ask them to modify the menu and they would refuse.
"One time a chef offered to prepare a special vegan curry for me and I just felt really bad about it. I just dreaded eating out.
"I'm not a massive fan of cooking myself so it was also time consuming."
Vegans eat no dairy products, eggs or anything derived from animals.
The Vegan Society told Newsbeat: "Some people, including vegetarians, may think being vegan is difficult but as with any lifestyle change, it simply takes some time getting used to it.
"You do need to set some time aside to do research on veganism but after a few weeks it will feel as natural as anything and you will start experiencing amazing benefits.
"Perhaps it's easier for vegetarians to see that eating meat results in animal suffering and death but the dairy and egg industries are not any different - all farmed animals are slaughtered at a young age and their welfare is compromised when profit is involved."