Mbappe, who picked up a hamstring injury on Feb. 1 and was expected to be ruled out of action for three weeks, will start on the bench against Bayern, sources told ESPN.
PSG have been in poor form recently, losing 3-1 at Monaco in Ligue 1 on Saturday for their fourth defeat in their last 11 matches. Bayern have not suffered defeat since September.
"Kylian trained with the squad today and completed the whole session. I didn't think he would be ready," PSG coach Christophe Galtier said at Monday's news conference.
"We took the decision to let him take part if he felt ready. We're not quite sure if he'll be on the team sheet, we'll see tomorrow morning.''
Galtier said he won't take any risks with Mbappe and he will have the last word about his presence against Bayern.
"We'll ask Kylian if he feels ready, Kylian's made a huge effort. If he's on the team sheet it means he's ready,'' Galtier said.
Neymar hopes to line up alongside Mbappe and Messi.
"Kylian is an extremely important player for us, he's a star. Obviously we feel stronger when all three of us are playing," Neymar said.
PSG have been in poor form recently, losing 3-1 at Monaco in Ligue 1 on Saturday for their fourth defeat of the year in all competitions.
Against Bayern, PSG will need to raise their level significantly.
"We have this ability (to switch to Champions League mode). I have a lot of belief in myself. I know what I'm capable of. I'm humble," said Neymar.
"It's an important game tomorrow and we're prepared for that clash. We have the team to play a great game tomorrow. I like to play these big games against big teams."
Information from Reuters and the Associated Press contributed to this report.