The next stop on the chain would be Salem, which is the Red Sox's Class A Advanced affiliate.
"Haven't made a decision on it," said general manager Ben Cherington. "There hasn't been a specific discussion about him. There tends to be a handful of players that get moved up toward the end of the year just to let them get a taste of the next level, so that's a conversation that we need to have with Ben Crockett and the player-development staff."
Overall, the Red Sox like what they've seen with Moncada, who got a late start after not signing with the club until March.
"I think it's been a good year," said Cherington. "I think we know that, with guys like this, that first year, there's a lot on their plate, dealing with a lot on the field, off the field. He didn't really have a real Spring Training.
"He started playing and I think he hit a little speed bump, but over the last 4-6 weeks he's been a very good player, one of the better players at that level for his age. He's doing a lot of different things on the field, running the bases.
"We don't think he's a finished product by any means. There's still aspects of his game that can still improve, need to improve, and he needs to play more in the Minor Leagues. But he's showing us everything that we thought he could do, at least in glimpses, and we think he can be a real good player moving forward."
Ian Browne is a reporter for MLB.com. Read his blog, Brownie Points, follow him on Twitter @IanMBrowne and listen to his podcast. This story was not subject to the approval of Major League Baseball or its clubs.