This is the first time that the social media website has issued such an alert, but companies like Google and Facebook have issued similar warnings in the past to their respective users.
"We believe that these actors (possibly associated with a government) may have been trying to obtain information such as email addresses, IP addresses and/or phone numbers," Twitter said in its notification, which some recipients then posted on their Twitter feeds.
It's unclear how many users were warned, but Twitter said that only a "small group of accounts" may have been targeted. Many users who acknowledged having received the message are privacy advocates and security researchers, some of whom tweet under pseudonyms.
Twitter referred potentially targeted users to the Tor Project and the Electronic Frontier Foundation for tips on how to protect their identities online.
Recommending Tor is a bit ironic, since Twitter was accused in the past of blocking users who accessed its website via Tor and forcing them to verify their accounts by providing phone numbers -- the same phone numbers that "state-sponsored actors" are now supposedly trying to access in order to identify activists who tweet anonymously.