The social media giant took the step Wednesday after hackers apparently hijacked the twitter account of actress Chloe Moretz, which tweeted out expletives before the tweets were deleted.
It came five days after a shadowy group called the "Chuckling Squad" took over Twitter CEO Jack Dorsey's account.
"We're taking this step because of vulnerabilities that need to be addressed by mobile carriers and our reliance on having a linked phone number for two-factor authentication (we're working on improving this)," Twitter announced in a tweet.
After closing up 1.4% on Wednesday, shares of Twitter fell 0.16% in after-hours trading to $43.28.
In another note, Twitter did not offer a specific timetable when it would restore the ability to tweet via text.
"We'll reactivate this in markets that depend on SMS for reliable communication soon while we work on our longer-term strategy for this feature," Twitter stated.