Lukic is the 19-year-old son of former Leeds and Arsenal 'keeper John Lukic.
Lukic jnr said on Grimsby's official website: "I'm pleased to sign for Grimsby and just want to be playing in the first team. He (Lukic snr) hasn't coached me but he's always been there for me, he comes to all my games. At Forest he was there at all my games and it was good. He has always been there for me to talk to and he has given me a lot of advise.
"He took me up to Barnsley a couple of times last year, that's where he has been working part-time. He has watched me in games and pulled me aside and told me things.
"I think it is the same how every goalkeeper starts, I was playing out field and one day I went in goal and did alright. I was playing in the local Sunday League when Forest took me in when I was eleven, I've been there ever since.
"I was third choice at Forest, behind Barry Roache and Paul Gerrard. My ambition is to have a half decent career and enjoy playing. It would be nice to follow in my dads footsteps."