Nowadays, the word is nowadays. You can find it in any dictionary (unlike now days). The better ones will also have the etymology:
late 14c., contracted from Middle English nou adayes (mid-14c.), from now + adayes "during the day," with adverbial genitive (see day).
As you can see, it used to be two words — seven centuries ago.
The Corpus of Contemporary American English does have a few cites for now days, but frankly, just look at the figures yourself:
now days 7
And here are the figures from the British National Corpus:
now days 0
That's how tiny a minority you're in. For once, the spellchecker is actually right.