Although representatives of the NYSE went calling on the microblogging site's San Francisco headquarters recently, sources familiar with Twitter's thinking say the conversation is ongoing. That would mean that a possible listing on the Nasdaq Stock Market remains in play.
Yet the mere suggestion of Twitter—one of the marquee names in technology—making its home at the NYSE may sting. While the Nasdaq is the destination of choice for most tech listings, including Facebook, the market has grappled with a string of high-profile blunders.
Facebook's 2012 IPO was widely considered to have been mishandled, and resulted in the Nasdaq being sued. Meanwhile, a massive outage last month left the stock market red-faced, and is still shrouded in controversy.
--Reporting by CNBC's Julia Boorstin; writing by Javier E. David