Twitter has no option to automatically import your Facebook friends and see their Twitter accounts at the time of publication. To do that, you'll need to use a workaround or resort to a more personal touch. The simplest option is to import your Facebook contacts into either a Yahoo or Outlook account and then import the contacts from Yahoo or Outlook into Twitter.
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Twitter has a "Find friends" option in its Discover section. Using this option, you can check several social media and email accounts. Twitter compares the email addresses in your contact lists with the email addresses of Twitter users and reports the account names of any matches. However, this feature doesn't let you directly check your Facebook account.
Yahoo or Outlook Solution
One way to automate the process of importing Facebook contacts to Twitter is to use a Yahoo or Outlook.com mail account. Set up an account on one of these services if you don't already have one and use the Facebook import tool in either Yahoo or Outlook to add your Facebook contacts to Yahoo or Outlook.com. Use Twitter's "Find Friends" feature to import the Yahoo or Outlook.com contacts, which now include your Facebook contacts.
Some Tips and Warnings
To simplify the process, log in to all three accounts -- Twitter, Facebook and either Yahoo or Outlook.com -- before beginning the import process. You'll find the Facebook import tool in Yahoo under the Contacts tab, accessible through an icon with a headshot silhouette. In Outlook.com, select the down arrow next to "Outlook.com" in the toolbar and select "People" to access the Facebook import tool. If you use the Outlook.com option, provide Microsoft permission to post to Facebook on your behalf when prompted, or the import won't work. You can withdraw this permission later by visiting Facebook and selecting "Settings," followed by "Privacy" and then "Apps" to review your permissions for individual services.
Using either Yahoo or Outlook as a workaround takes a little while, and the results aren't guaranteed. Even if everything goes as planned, you won't get contact details for anyone who used a different email address to register for Facebook or Twitter. To improve your chances of finding more of your Facebook friends on Twitter, post a status update on Facebook asking your friends to share their Twitter usernames, either in a reply to the post or a Facebook message.