A spokesperson for the social networking site confirmed to The Next Web that it is evaluating the potential new function with "a very small percentage" of users.
Twitter users @psilosophy and @nspeaks told the website that the function has been made available to them via the Settings menu on their accounts.
If they select the relevant option in their settings menu, users are sent an email inviting them to download their personal archives.
As it currently stands, the "download all tweets" tool presents users' archives as a .zip file which, when extracted, is an HTML archive that allows browsing by calendar month.
The archive is also available in .csv and .json formats, and users can search through all of their tweets.
Twitter launched its own integrated photo filters last week, following Instagram's decision to end its Twitter support earlier in December.