Many of the people to receive the emails appear to work or be active in internet security, particularly related to Tor, the browser used to access the dark web. Twitter did not say which country may have been attempting to attack users, although the usual suspects are China, Russia, North Korea and America.
Users reporting that they had received the alerts included Coldhak, a Canadian non-profit organisation which says it is "dedicated to furthering privacy, security and freedom of speech"; Noris Fabio, a security researcher; and Runa Sandvik, a security adviser.
Twitter said they had been sent to a "small group of accounts", and that it was investigating the issue further to see whether any data had actually been taken. It said that if users wanted to better protect themself online, they could use anonymous browser Tor, and also directed them to theElectronic Frontier Foundation.
"It's possible your account may not have been an intended target of the suspected activity, but we wanted to alert you as soon as possible," Twitter said in the warning.
The alerts follow similar steps taken by Google in 2012 and Facebook earlier this year.
Many activists or other users who tweet using a pseudonym to hide their true identity may be targeted so that hackers can find out who they are. Email addresses and phone numbers may then be used to gain access to another internet service.
Facebook has said that government-sponsored attacks tend to be "more advanced and dangerous than others", while Google says it uses reports from victims to help determine if a state is attempting to hack them.