Edit - Now this is the top answer, I'll quickly answer the question, which is yes, you certainly can! But please read on...
I'd like to put forward a caveat - even if your teacher is willing, this may not be your only concern. It sounds like you might be interested in learning a very different skill. Let me explain through a quick personal interlude.
I was roughly classical grade 7-8 playing standard, and I asked my teacher if we could move towards jazz and 'modern' music, and he happily obliged. It turned out he was very talented at playing 'by ear', and so he basically taught me that skill from scratch. My finger dexterity was useful, but everything else I had to relearn, and this included not being dependent on sheet music at all. There are many new skills involved with this including knowing more scales, understanding chord theory, 'hearing' chord progressions, etc... and of course a lot of practice!
I was lucky that my teacher could teach these skills. If he hadn't, he probably would have tried to get me to find sheet music of the songs and teach me in the same way that most teachers are used to teaching. This may be what you're after, in which case no problem. If you want to embrace a very different form of piano, however, it may be possible that you will need a different teacher regardless of willingness since even some excellent sheet music teachers may not have the skills to teach this very different skill.
After 20 years I have now re-embraced classical (you can't beat playing a bit of Beethoven...) but I continue to write my own 'modern' pieces, improvise, and play along to songs I like as well. Additionally I can sit down at the piano and jam in any key with other musicians, or get someone to sing a song and play along with them after just a few run throughs to 'get' the chords on which I can then play the tune or improvise a backing. This more spontaneous, social aspect of music has bought me an enormous amount of enjoyment.
To be honest, I don't even think I'm that good! I don't practice nearly enough, and am ill-disciplined with my practice (maybe I should get a teacher again!) However my experience is that it's a much less common skill than being able to read music. However it is not an easy transition; beyond the obvious interface between hands and keyboard, I find the two to be almost completely different skills.