Odegaard, who does not turn 16 until next week, has been attracting interest from the elite of European football after starring for Norwegian side Stromsgodset
The attacking midfielder became the youngest player to represent Norway earlier this year and his impressive displays have seen him linked with the likes of Liverpool, Manchester United, Arsenal and Real Madrid.
The player himself reportedly spoke of his "dream" to represent Liverpool last month, but it is not yet clear who will win the race for his signature.
Bayern chairman Rummenigge is confident that the youngster will opt to join Pep Guardiola's men after spending time with the club earlier this week.
"We will try very hard to sign him and I think we have a good chance," Rummenigge told Bild.
"He is an absolutely exceptional talent and a good guy, very mature for his age."
Odegaard made 23 appearances in Norway's top flight in the 2014 season, scoring five goals.
