It’s a good book, but if the 1800s is too early for you, Jonathan Strange isn’t going to be what you’re looking for.
Another great book involving magical apprenticeship, though it may not be modern enough for you, is Lord Dunsany’s The Charwoman’s Shadow.
With a Single Spell by Lawrence Watt Evans is an apprentice story, although an apprenticeship cut short.
Ah, missed the “set in modern times bit”, since Ethshar isn’t exactly Earth circa now.
The Iron Druid series (author Kevin Hearne) has a master-apprentice relationship, but I’ve only read the first book, so I don’t know if it gets sexual/relationshippy in the later ones.
The first book (Hounded) does have some nice detailed accounts of the apprenticeship training aspects, and the difficulty of learning to work magic, and as I said - no relationship yet.
Well, came in to recommend the Dresden and Iron Druid series, but others already recommended them so I’ll just second or third them. Both are excellent IMHO. I love the unabridged audio of the Iron Druid, as the dog is just hilarious.
Robert Rankin’s “Brightonomicon”.
Can’t believe I forgot that one :smack: And nope, not even a threat of any kissy-kissy for the protagonist and his apprentice throughout the rest of the series.
Much of the Chrestomanci series by Diana Wynne Jones (beginning with Charmed Life) is centered on magical apprenticeship. There’s also a parallel worlds theme.
Cool, thanks! I’m a little overloaded now (I picked up several of the Butcher books, “The Magicians,” and two of the “Peter Grant” series) so once I work my way through those I’ll check back and see about some of the others. 
Another book that might be good is The Dark is Rising by Susan Cooper (part of a series of the same name). It was written in the 70’s but set in then-current times, and draws on a lot of celtic myth.
I immediately thought of this series. It’s definitely all about the magic apprenticeship. And arguably it’s in modern times - clearly Aahz is familiar with modern Earth - it’s just that it’s set in other dimensions.
The Bedlam’s Bard series by Mercedes Lackey (and various co-writers) has some of this. The first book is about the protagonist discovering that magic exists and he’s a Bard, then in later books he becomes an apprentice to a Master Bard, gains experience and eventually takes on an apprentice of his own. There’s also a subplot with a Healer and her apprentice. There is some romance, but none between master and apprentice.
I say this in the nicest possible way, but those books are SOOOOOOO dated! I loved them when they first came out, and I re-read them recently, and still enjoyed them, but oh man… so very very dated.
But yes, enjoyable, good characters, good variety of relationships, and a decent bit of the “teaching/learning” process. Not all of the involved master/apprentice characters are HUMAN, if that makes a difference to the OP. (Some are elves)
While I wouldn’t consider the Iron Druid Chronicles as romance by any means, I think “not even the threat of kissy-kissy” is going a bit far.