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The offer came just days after he left his role at Fleetwood Town, but he was concerned about the club's position and finances.
Brown said on The Warm-Up with Kris Boyd and Chris Sutton: "We left Fleetwood and two days later, I'd driven my stuff back up the road, and Cheltenham phoned.
"I've driven myself to the front door, walked into the stadium, had the meeting, everything was fine and literally, there and then, they've offered me the job."
However, despite being the favourite for the job, Brown decided not to take the role. He explained: "They were struggling at that point in time; they were second bottom of the league, they were 15 games into the season.
"They'd already admitted the money wasn't there; they're happy enough to go down; it would be a rebuild the next year."
The former Celtic player eventually found his way back to the dugout with Ayr United but has not taken on a managerial role since leaving the Championship club.
And while Brown admits he misses the daily routine of coaching, a return does not appear imminent. He said: "I definitely do miss it.
"I probably will go back in at some point in time, but I'm not sure it's going to be any time soon.
"As long as I'm part of something, I enjoy what I do just now."