I am sorry if the question is silly, but I think I heard both options spoken by native speaker. So, which one is correct?
I agree with Steve Melnikoff that put something in sounds more natural. Put something into puts the emphasis on the motion, put something in on the target.
Examples for use of put something into that portray that fact are:
- put something into motion (formerly motionless)
- put into play
- put into effect
There is a verb put in something, meaning to apply, plant, spend (time/money), introduce, (nautically) enter a harbor or port.