Re-watching "The Big Goodbye" last night (excellent episode) and having watched the episodes in order before it to this point, I had forgotten how much they pushed the Picard-Beverly sexual tension so much early on. Unless I'm misinterpreting the intent, Picard initially invited Beverly (and only Beverly) to join him on the holodeck almost as a "date" or in a romantic sense, and it was subsequently suggested to him that he bring Whalen. Later in the episode, Beverly asks to see his office (which was preceeded by their first encounter at the police station where it looks like they're going to kiss) and they both look frustrated when Whalen and Data request to come along.
Add this up with EAF, The Naked Now, and at least one other I can't recall all the moment, and it really seemed like they were heading in that direction with the relationship.
Interesting to think what might have been, and really until "Attached" and "AGT" years later (with the occasional breakfasts), it was not even so much as addressed for most of the series after being ever-present early on.