The short answer, as per Betteridge's Law, is no.
The slightly longer answer, as per common sense, is definitely no permanent transfer, but obviously we'll be looking to send him out on loan to challenge him beyond the level that the U21 Premier League and UEFA Youth League offer. Musonda himself hinted at as much, too, back in May, back when Monaco and Anderlecht were rumored as potential loan destinations.
"Everything will depend on the discussions that I will have with the coach [Jose Mourinho] and the club's management at the end of the season. [...] My priority is Chelsea and, obviously, if I can get more playing time, I will stay. But if this is not the case, I will ask to go on loan."
-Charly Musonda, source: Foot Mercato via Evening Standard
Presumably these discussion have been had by now -- everybody's on vacation currently! -- especially since Musonda did join the team on our post-season tour of Thailand and Australia. While Anderlecht seem to have dropped out of the reckoning, Monaco remain (possibly as part of a Falcao deal), and have been joined by the likes of Marseille (who were also looking at taking Jeremie Boga on loan), and PSV (who are down one Memphis Depay for next season). Any of those three options would fit the bill as the next step in Charly Jr's development.
This is all quite reasonable so far, but recently, the 18-year-old has been linked with permanent moves away from Stamford Bridge, too. Citing quotes from an unnamed "source close to the player," the story goes that Musonda is suddenly not happy with the prospect of a loan and wants immediate, meaningful first-team action at a top club in Europe.
"Charly wants to play at the highest level next season and he wants to go to a club where he can have guarantees in terms of playing time, and who will be of Chelsea's calibre."
"With the Blues, he does not have guarantees, and that disturbs him, especially as he is very attached to the club."
Even if we entertain this as a serious notion for a moment, it's hard not to laugh when the list of potential suitors are drawn up.
That's right, apparently City, Arsenal, and PSG are just absolutely champing at the bit to throw down €10-15m and plug Musonda into their starting lineups (with guarantees!) as they look to challenge for Premier League, Ligue 1, and Champions League glories. Sorry, Mr. Anonymous Source, that's just a touch too ridiculous, regardless of how talented Musonda may be.
Charly Jr recently signed a new contract, tying him down to Chelsea for the next four seasons through 2019. The time to cash in isn't anytime soon.