The Bundesliga veteran claimed the English starlet, 18, needs to take the "big step" in order to achieve his dreams.
Bayern have filed a fourth offer of £35million for the forward, who is under contract with Chelsea until 2020.
Robben, who has played for them since 2009, has advised him to take the opportunity with both hands.
Speaking to beIN Sports, the 34-year-old said: "He is one of the best [young players] in Europe.
"You are better off getting your chance and if you get that chance then you have to take it.
"For young players, it's always the same. First, you have to take the big step to achieve it.
"Become a professional football player at the highest level, get into the first team, get your place.
"Then, second, which can be more important, what's sometimes missing is that you have to stay there.
"Staying there is often more difficult than actually becoming a footballer in the first place.
"You have to prove it, game after game, season after season and don't be happy.
"Don't just be happy when you achieve something little or small."
SunSport exclusively revealed that Bayern Munich plan to give Hudson-Odoi the iconic No10 shirt from Robben next season.
But Chelsea boss Maurizio Sarri has lashed out at the Bundesliga giants and called their approach of his player as "unprofessional."
He said: "I think it is not professional. They are talking about a player under contract with Chelsea, so they didn't respect our club I think."
Chelsea star Marcos Alonso has urged his team-mate to reject the Bayern offer and fight for place at Stamford Bridge.
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