There have been some rumors and confusion about a possible Alien vs Predator 3. Some confusion is due to the title because there has already been an Aliens vs Predator 3 video
game (although it was mostly marketed as Aliens vs Predator). We take a short look at the speculation for the movie sequel and also in the video game.
Also, note that in some cases Alien is used plurally (Aliens) and sometimes singluarly (Alien) in the franchise titles.
The Aliens vs Predator: Requiem movie (also known as Alien vs Predator 2) left the door open for a sequel. In the story, a woman named Yutani was introduced and hinted a Weyland and Yutani merger process. Also, several main characters were left alive at the end of the movie (although the lead Predator died). One of the best things in the movie was seeing the Predator homeworld (see picture below) which could also be included in the new movie. The directors of the movie - the Strauss brothers have shown interest in making the sequel and making in the future, with the action happening in space (this would enable them to use Colonial Marines as well). Aliens vs Predator: Requiem did not do well critically or financially, so the idea is mostly buried. We can be sure that Tom Woodruff Jr. and Alec Gillis would be involved with Alien vs Predator 3 if ever made. Ridley Scott and James Cameron have expressed their dislike of the AvP series in several interviews. As long as Ridley Scott is still active with the series, it will hinder the possibility of Alien vs Predator 3 happening.
The Hunt: Alien vs Predator was a 1991 script draft written by Peter Briggs (one of the writers of the first Hellboy movie). Briggs was the first person to undertake the project, although after his script was rejected development of the film would stall for almost a decade until Paul W. S. Anderson picked it up. His script bears no relation to Alien vs Predator as it was ultimately made, and is instead based more closely on the original Aliens vs Predator comic book by Dark Horse Comics written in 1990. The script is very good and would be a great basis for the possible Alien vs Predator 3 movie. It is set in the future and finally includes the Colonial Marines. It is available to read in HorrorLair.
In 2010, Sega released a sequel to Aliens vs Predator 2, titled just Aliens vs Predator. Rebellion, the developers of the first game in the series, returned to develop it. The action takes place on a jungle planet named BG-386. There is a secret research facility run by Carl Bishop Weyland, who is actually an android based on Peter Weyland, the character from the first AvP movie. Lance Hendriksen also returns to give a voice to him. The ending of the game hints heavily on presence of a Xenomorph Hiveworld (origin planet of the Aliens). Of course, the Predators are interested in going hunting to the planet and the humans are interested in the research and bioweapons possibilities. The planet is seen on a Predator ship scanner and seems to be lava and rock-based planet, similar to LV-426. The game is heavily influenced by the 2 AvP movies and can be considered Alien vs Predator 3 in game form.