According to the Stuff website, the 28-year-old two-Test All Black injured his foot in Friday's 27-21 defeat to the Chiefs at Eden Park that put the Blues into an early 0-2 hole for the new Super Rugby season.
Pulu's absence will leave the vastly less experienced Sam Nock and Jonathan Ruru to share No 9 duties in clashes against the Lions and Stormers.
Meanwhile, there was some welcome good news for under-pressure coach Tana Umaga with All Blacks star Sonny Bill Williams, who missed the Chiefs match with knee and head knocks, training fully and confirming his participation.
The Blues will play the Lions (Saturday, March 10 at 17:15) and Stormers (Saturday, March 17 at 15:05) before enjoying their first bye of this year's tournament.
After the bye, the Blues will return to Eden Park to take on the Sharks in Round 7.