The Last of Us Part 3 doesn't exist but this hasn't stopped fans from asking for the game. It also certainly doesn't help that insiders keep talking about the game as if it's already in development. We know from Neil Druckmann the outline exists already as he's mentioned it several times in the past, including earlier this year. But, leakers insist that it's in development.

Right around the time the May 24 PlayStation Showcase took place, @RPKGaming1 sent The Last of Us fans into a frenzy when it claimed that The Last of Us Part 3 "is currently in the works."
Naturally, fans couldn't help but contain their excitement. Some even went as far as to theorize that it could be announced soon.
Nevertheless, no official confirmation has happened since the game went viral, which leaves us with the question, is The Last of Us Part 3 really coming?
Well, if you believe the source itself, many have pointed out that they're more solid when it comes to movie leaks. But, as far as gaming leaks go, their track record is spotty. Don't forget, they're the ones that claimed that NetherRealm Studios was working on a Marvel fighting game, which has since been denied by Ed Boon.
But, realistically speaking, it's impossible that Naughty Dog and PlayStation Studios haven't at least started pre-production on The Last of Us Part 3.
Having an outline is one thing but getting the ball rolling on a threequel for one of the most popular and awarded games of all time is an entirely different story. It's perfectly understandable if Naughty Dog wanted to call it quits after The Last of Us Part II. However, the success of The Last of Us on HBO, which is getting multiple more seasons, confirms that there's a fervent demand for more stories in the post-apocalyptic world inhabited by the likes of Joel and Ellie.
Given how long and how expensive AAA games take these days, Naughty Dog should start working on The Last of Us Part 3 this year so the game can be released by the time that the third season of HBO's live-action adaptation comes out.
The production on Season 2 of The Last of Us is currently paused following the writer's strike, which could delay it from its current release window. In turn, this will likely affect the release of Season 3.
In all likelihood, Seasons 2 and 3 of The Last of Us will be out by 2025 and 2027, respectively.
Four years is plenty of time to get The Last of Us Part 3 made if Naughty Dog poured all of its resources and started from scratch now.
Ultimately, this is all speculation and there's no official confirmation that The Last of Us Part 3 is happening. But, the good news is that a multiplayer spin-off is coming. While it wasn't revealed at the May 24 PlayStation Showcase, fans might hear more about it on or before September 26, which is celebrated as The Last of Us Day by Naughty Dog and its fans.