The 27-year-old has been sidelined since February 12 with an abductor problem which required surgery.
Celtic face Brown's former club in the Hampden final on May 26 and the Scotland midfielder is hoping to make the showpiece occasion in which the Hoops will have Victor Wanyama and Beram Kayal missing from the engine room 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," he 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."
Celtic need only a point from Sunday's home game against Inverness to wrap up the Clydesdale Bank Premier Premier League title, although second-placed Motherwell could spoil the party if they fail to win at Dundee United on Friday night.
Brown said: "It would be a little disappointing if they [Motherwell] get beat and it's already over by Sunday. Here's hoping they can do it and we can win it on Sunday. I would rather do that."
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- Victor Wanyama