(FORTUNE Magazine) – TIME WARNER CEO Dick Parsons, the most reserved (some would say inscrutable) of media moguls, must be struggling to suppress a self-satisfied grin right about now. His company's online unit, AOL, has suddenly become the hottest property around, entertaining suitors such as Google, Comcast, Microsoft, and Yahoo, all of which seem willing to pay a premium price for a stake in the business. (Time Warner isn't commenting.)

I-told-you-so's notwithstanding, it's probably the wrong time for Parsons to sell a piece of AOL. The most recent buzz is that Time Warner (FORTUNE'S parent) has been looking at offers that would value AOL at $20 billion--roughly double its current Wall Street valuation but well below its potential. Ken Marlin, managing partner of the boutique investment bank Marlin & Associates, figures AOL could boost its annual ad revenue from more than $1 billion to $6 billion by 2009. That could translate into a market value of $50 billion or more. And AOL just launched a couple of online voice products-- another value booster when you consider eBay is paying up to $4 billion to acquire Skype. Besides, with fellow moguls like Rupert Murdoch hungrily eyeing the web, it may also be the wrong time for Parsons to enter a complex deal that could distract Time Warner from using AOL to distribute its traditional media holdings.

Investor Carl Icahn recently accused Parsons and Time Warner management of "habitual excess deliberation." In the case of AOL, habitual excess deliberation may just work to Parsons' advantage.