I have this set. It leaves a 1/4 inch border when you use the cut-out part.
https://www.katscrappiness.com/colle...at-scrappiness
Her self-branded dies are a really good price. And what I like about this set is you can make stitched frames with them when you use 2 dies. I also have this set
https://wholesalecraftingsupplies.co...-marie-designs but the border size of the second die is probably also 1/4 inch or slightly more. She also does great prices on her dies. The only thing is the shipping is high - probably to encourage more sales. Though she ships free if you spend over $50. (it's worded over $50 - not sure if it's actually at $50 since it's been awhile and I can't remember.)
If you're looking for a thinner border than 1/4 inch (like 1/8th, I don't know if there are any. I think most probably start at full card front size and then go down in 1/4 inch.