A Twitter spokesperson confirmed that there is a "technical issue with URLs in direct messages," but did not provide any additional information as to what caused the problem.
Twitter Says A ?Technical Issue? Is Preventing URLs From Being Sent In Direct Messages For Some.
Twitter has posted a message on its DM help article that reads:
We're restructuring back-end elements of our direct message system. As a result, users may be unable to send some URLs in direct messages. We apologize for the inconvenience.
Source: Twitter Support
It turns out that the technical issue may have been referring to the handling of Direct Message spam, not with the sending of URLs. A wave of hacked accounts a few weeks ago has led to a massive rise in DM spam, where links are sent to users in the hopes that they will click and enter personal information to be collected by scammers.
Links go through without a problem for www.twitter.com, www.instagram.com, www.google.com, and www.facebook.com, so it appears Twitter has unblocked links from specific websites. This is consistent with the idea that the link ban is a bandaid fix to combat a problem with malicious links, since the service is still allowing links to websites it knows to be safe