MaraxuS80's estimate seems pretty spot on. Also keep in mind that they might adjust season length depending on if they want it closer to Blizzcon - or rather have S13 stuff ready on PTR for after Blizzcon, so this season can be potentially longer or shorter as usual. But I cannot imagine anything before end of September as Don Vu said in a bluepost they have a new balance patch for PTR that'll be deployed on September 10 or so, therefore I think a slightly longer S11 and a S12 that is still "fresh" during Blizzcon isn't entirely unlikely either.
We don't know exactly, but we can extrapolate from previous seasons:
http://d3resource.com/seasons/
MaraxuS80's estimate seems pretty spot on. Also keep in mind that they might adjust season length depending on if they want it closer to Blizzcon - or rather have S13 stuff ready on PTR for after Blizzcon, so this season can be potentially longer or shorter as usual. But I cannot imagine anything before end of September as Don Vu said in a bluepost they have a new balance patch for PTR that'll be deployed on September 10 or so, therefore I think a slightly longer S11 and a S12 that is still "fresh" during Blizzcon isn't entirely unlikely either.