Top seed Ivanovic, whose match could not be completed on Tuesday, was serving at 2-1 down in the third set against Ayumi Morita when play was stopped, while twice defending champion and sixth seed Daniela Hantuchova was 6-2 1-0 up on Olga Puchkova 6-2 1-0.
Fellow seeds Sabine Lisicki and Elena Vesnina were scheduled to be in action later in the day.
Three first-round matches had already been held over from Tuesday due to the weather.
With an improved forecast for Thursday, those matches should be completed but the winners will have to play their second-round matches later in the day.
The quarter-finals are planned for Friday, the semi-finals Saturday and final Sunday.