The Bundesliga giants tried to sign the 18-year-old from the Blues in January , seeing several bids turned down.
Hudson-Odoi, who had grown frustrated with his lack of playing time in London, handed in a transfer request to force a move , but the London club would not let him go, while Bayern CEO Karl-Heinz Rummenigge said the German side will return with another offer in the next transfer window .
The Chelsea academy graduate may be more willing to stay beyond this season, but Elber has warned Bayern that they must not give their European rivals a chance to snap him up ahead of them.
"We can only know if Hudson-Odoi is suitable for Bayern after his signing," Elber told Goal. "However, Bayern need to be cautious when signing new players. The current transfer market is indeed the case, €5 million cannot buy the players you want.
"If Bayern don't make the call, maybe Real Madrid and Barcelona will."
Bayern returned to the top of the Bundesliga on Saturday after beating Borussia Dortmund 5-0 to overtake their fierce rivals.
With just six games to go, the result could prove crucial in the battle for the German crown and Elber was happy with his former side's performance.
"Before the game, Bayern Munich were under a lot of pressure and was two points behind," he added. "We had to win this game. Bayern Munich showed their desire to win from the beginning of the game, to force Dortmund to make mistakes. And Bayern did seize the opportunity."
James Rodriguez was an unused substitute against Dortmund despite his fine form in the Bundesliga. The Colombia international's two-year loan from Real Madrid will come to an end this summer, but Bayern have the option to buy him on a permanent basis.
With seven goals and four assists to his name this season, Elber expects James to still be in Bavaria next term.
"James didn't get a chance to play against Dortmund, but he has played a lot in the latter half of the season," he said. "Bayern need quality players, and James is a quality one. I think he will remain in Bayern's squad next season."