Davies, still just 19, joined Bayern from Vancouver Whitecaps in January 2019 but has forced his way into Hansi Flick's first team this season.
The speedy Canada winger has forged a starting role at left-back, meaning David Alaba has played in the centre of defence while Lucas Hernandez has spent much of his first Bundesliga campaign injured or on the bench.
Davies was named the German top flight's Rookie of the Season and particularly excelled in both legs of the Champions League last-16 tie with Chelsea.
Now, in the final eight in Lisbon, Davies might be tasked with handling six-time Ballon d'Or winner Messi.
But Rummenigge said: "Alphonso had a great season and was rightly voted Rookie of the Season.
"Should Messi play on his left side, Alphonso will take care of him. It will be a difficult but interesting task. Davies hasn't been dribbled past or overrun much this season."
Davies is certainly relishing the prospect of taking on his boyhood idol, telling UEFA.com: "Growing up, I used to watch him all the time and now I'm defending against him; it's going to be a whole different ball game.
"But for me, it's just playing my game, changing nothing on my side and just hopefully doing my best against him.
"We know that he's a great player - we're not going to take that away from him - so the best you can do is try, I guess."