The 34-year-old has previously said that he will decide his future in July, with many believing that he has played his last game for the national side.
However, Schmidt, whom O'Driscoll said last week taught him more in three years than he learned from any other coach in the previous decade, is hoping that O'Driscoll decides to stay on.
"He'll make his own decision but at the same time it doesn't stop you trying to push him towards one," Schmidt told Sky Sports News.
"I'd like to see him continue, I'd love to see him named tomorrow [in the British and Irish Lions squad], I think that would be part of a continuation of a stellar career, hopefully not the end of one."
Schmidt was named as the new Ireland boss on Monday.