The 29-year-old De Bruyne sustained the injuries in a collision with Chelsea defender Antonio Rudiger while playing for Manchester City in the Champions League final on Saturday. De Bruyne had to be replaced in the 60th minute of the match, which Chelsea won 1-0.
Martinez told a news conference from Belgium's training camp that the prognosis for De Bruyne was encouraging and that the playmaker will link up with the squad after a few days' rest.
Martinez said he was not sure if De Bruyne would be ready for the team's first match of Euro 2020, against Russia on June 12.
The Belgian FA is expecting De Bruyne to be with the squad by next Monday, but only after taking tests to check his injuries are healing OK.
"Kevin had a different programme to the rest of the players, he was only going to join us seven days later because he was playing in the Champions League final," Martinez said.
"But that programme has obviously been changed, he won't be able to relax and rest just yet."
Martinez said De Bruyne faced a series of hospital tests, returning first to Manchester before joining up with the Belgian side.
"We've got to be very cautious, and we will get more clarity in the next days, but I've spoken to Kevin and he was quite positive," the Spanish coach added.
"We are very fortunate that even though he has a double fracture, he doesn't need surgery. An operation would have made it impossible for him to play in the European Championship."