Twitter did not immediately respond to a request for comment.
Shares of the company jumped as much as 4 percent following the note and closed 3.6 percent up at $38.43 per share. Twitter shares have been on a prolonged slide for the past few months, dropping near its 52-week low of $29.51.
Costolo has been taking heat after a filing last week revealed his family trusts sold approximately $5.32 million in Twitter stock; the trusts, named for Costolo and his wife, no longer hold any Twitter stock, Bloomberg reported.
There have also been serious questions about how well suited Costolo is to his current position. The executive has had the difficult task of growing and monetizing Twitter -- a job that's become even more of a pressure-cooker since the company went public last year. Despite a redesign and many new ad products, the company has yet to show a clear path to making money off advertisements.
Meanwhile, Wall Street analysts have raised some concerns over the slowing pace of Twitter's growth. In its October earnings report, Twitter reported quarter-to-quarter growth of 4.8 percent, down from 6 percent growth in its previous quarter.
The company has 284 million monthly active users, and was recently passed in size by Instagram.