The Darwath Trilogy by Barbara Hambly. It’s set within the past 20 years or so, but a couple of the characters cross over from our world into a parallel world, where magic works and various prehistoric critters are not extinct. There’s some romance in the books, but not between mage and apprentice. One of the sequels, Mother of Winter, shows the second generation of apprenticeship, where the former apprentice takes on an apprentice of his own, and gets payback.
Also, Jim Butcher’s Harry Dresden series has the main character take on an apprentice. The series is set in this world, in modern times. I slogged through the first book, and I think that Butcher’s writing has become better.