The Hoops captain has been sidelined since Febuary with an abductor problem, but has confirmed he is targeting the Hampden final.
This will be a boost for Neil Lennon's side who will be missing Victor Wanyama and Beram Kayal through suspension.
"I'm still a wee bit from training with the first team but, if everything goes well, hopefully I'll be back for the cup final," Brown said.
"That's my target but it will be touch and go. It's going to be hard to get some game time in before it as well.
"I'm feeling quite good, I've been back running.
"It's been four weeks since the operation now and I will take things slowly.
"I know we have a couple of lads out, but we have a big squad that can deal with that.
"It will be the manager's decision. He will see me in training in the weeks leading up to it.
"I would need to be honest as it would take a place away from someone who is 100 per cent fit, especially with me not playing in so long.
"So the next couple of weeks will be huge for me to see if I can make that push for the cup final."