Bond producer Barbara Broccoli has apparently got her eyes set on the 33-year-old for the next 007 adventure, The Sun reports -- which would bring an end to the super long wait to find the seventh actor to take on the iconic role since 1962.

So, here's the deal ...the publication reports nothing's set in stone yet, but word on the street is that Bond is Aaron's gig if he wants it. The formal offer is sitting right there on the table, with producers waiting to hear back from him.

But, Eon, the production company behind JB, is pretty darn sure that Aaron's gonna scribble his name on that contract any day now.

Just last week, Aaron was keeping it low-key when folks started buzzing about him playing Bond, telling Numero, "I find it charming and wonderful that people see me in that role. I take it as a great compliment."

Classy move. Keeping things cool until everything's official ... very Bond.

IYDK, Aaron's last film was "Bullet Train" alongside Brad Pitt, and he'll be appearing in 3 films this year, "The Fall Guy," "Nosferatu" and "Kraven the Hunter" ... the fifth film in Sony's Spider-Man Universe -- so he's been keeping busy, but looks like he'll soon be adding a whole lot more to his plate.

Goes without saying, the quest to find the next 007 has been one heck of a ride, with some seriously heavy hitters in the mix, such as Idris Elba, Tom Hardy, and Henry Cavill ... just to name a few. But, as for now, it looks like it's ATJ's role to lose.