McClean has broken onto the scene under Martin O'Neill and has put in a host of impressive performances, which has led to O'Shea to call on Ireland boss Giovanni Trapattoni to include the midfielder in the squad.
"I hope he gets in. He is going the right way about it. He is playing fantastic, creating chances and if we can score a couple of more goals before the end of the season then he will catch the manager's eye. He has a great chance," said O'Shea.
"It's been an incredible rise. It's dream stuff. Credit to him because you can easily have one or two good games in the Premier League and then become an unknown. He has become one of our best players though. Physically and mentally he has been able to cope.
"He will not be overawed if he did go to the Euros. He has shown that. He will take everything in his stride. The manager has got a squad for Ireland that has got us to the finals, so it will be a difficult decision to make. Whether he makes a couple of changes we will see."
Sunderland take on Bolton Wanderers on Saturday.