I played this song to my Korean friend. She agreed with Taegyung's comment - that she heard '만나는'.
I then tried saying the phrase myself to her, saying something more like 만-난-ㄴ 날, 're-voicing' the last 'ㄴ' so that it sounded like 3 syllables but without 'opening my tongue' again (so that I wasn't clearly enunciating that last '는'). That's what the song sounds most like to me.
I asked my friend what she thought I'd said - and she thought I'd said '만나는 날'. I tried again once to be sure, but she was definitely happy to interpret my '만-난-ㄴ 날' as '만나는 날'.
So it seems that you're right - in this instance, being a bit lazy with clarity of enunciation seems to be forgivable! I'm not sure if there's any specific Korean language rule that is in play here, or if it's just that when people are speaking fast in any language, some less significant sounds and syllables can be 'sacrificed' a little.