The Overwatch blog has details on changes Blizzard is planning for Battle.net, which will include the addition of voice chat. This will initially be tested in the Overwatch beta, but they indicate this will eventually be part of the Battle.net app:

We're in the process of working in some slight updates to Battle.net in preparation for the upcoming Overwatch beta test, and wanted to give you the heads up. This update will lay the foundation for us to test voice chat functionality within the Overwatch beta.

We're excited for players to give this feature a try during the Overwatch beta and look forward to your feedback. Further details on our voice chat plans for Battle.net will be released further down the road, but for now, please see the FAQ below.