Exactly that - a song is "on the verge" of making the top 40. Ryan has featured songs in this category for many years now. Kind of like Rick Dees' "Sure Shot" that he begins his shows with.
OTV was previously called "Out of the Box" and then "Breakout".
Actually they still have "Breakouts". I think the Breakout slots are songs that were recently released to radio while OTV songs are probably right outside the top 40 and would debut within a week or two.
Regarding one "On The Verge" - the song "We Don't Talk About Bruno" by the cast of Disney's Encanto has been in the OTV list for about 3 weeks when it has yet to make the Ryan Seacrest AT40 reaches. If it doesn't make Ryan's AT40 chart soon it may or may not be a contender to end up in the 2022 AT40 year end countdown by surprise just like "Ever the Same" by Rob Thomas did in the 2006 AT40 year end countdown with a few weeks as an AT40 Breakout song without making Ryan's AT40 chart at all.