He has been ruled out of Leinster's Amlin Challenge Cup quarter-final against Wasps on April 5.
And is unlikely to play again before Warren Gatland names his squad to tour Down Under.
"We have been given a five to six-week time frame," said Leinster coach Joe Schmidt.
"Whatever we are planning over the next five or six weeks will mean it's (Wasps) probably a little too soon for Jonny."
Sexton completed just one game during the Championship after pulling his hamstring against England in Round Two.
He was poised to return in Ireland's defeat to Italy but damaged a tendon in his foot while training.