Some fans are hoping the role, which was previously played by the late Carrie Fisher, will be continued with a new actress rather than be written out of any upcoming films.
The petition describes Streep as an "ideal candidate" who "has always been known as one of the greatest actresses alive".
"With three Oscar wins and 20 nominations (probably 21 by next year) her achievement in acting has long been recognised," it continued. "Apart from her acclaimed acting performances, her close relationship with Carrie Fisher and her performance in Postcards From the Edge also made her an ideal candidate to play Leia in Star Wars Episode IX."
"As fans of Star Wars and Carrie fisher, we really want Leia to shine in Episode IX and we certainly do not want her to be written out of the film abruptly without a reasonable plot," they continued. "Therefore, recasting Leia is a more ideal option for us and we believe that Meryl Streep is an ideal candidate to play Leia."
Postcards From The Edge was written by Fisher and based on her "semi-autobiographical novel". Streep played the character of Suzanne Vale - an actress recovering from substance abuse. A friend of Fisher's, she reportedly sang with Fisher's daughter at the memorial service held at her home after she died.
At the time of writing, the petition had 8,573 supporters out of a goal of 9,000. Many fans commenting on the petition expressed their support for the idea, with one writing that Streep "would do justice to Leia".
Another said it would be "disrespectful" to replace Fisher in a role that she made iconic. Many more said it was important for Leia's story to be told as she is "an important character for so many people" and "deserves her ending".
Online petitions about Star Wars are nothing new and so far have appeared to have zero influence on the multi-billion dollar franchise.
Fans will have to wait and see how Fisher's death will be dealt with in the final episode of the saga; LucasFilm president Kathleen Kennedy previously said the filmmakers had to "start over" after she died, as she was reportedly a huge part of the original script.
LucasFilm has said Fisher will not be brought back as Leia using CGI, despite the production company using the technique with Peter Cushing for Rogue One.
"We want to assure our fans that LucasFilm has no plans to digitally recreate Carrie Fisher's performance as Princess or General Leia Organa," they said in a statement.
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