There is nothing wrong. What you describe is exactly what is supposed to happen. To demonstrate what is happening, try the following exercise:
In Illustrator change your General Units in the Preferences to "Inches".
Place your 96ppi image
900px / 96ppi = 9.375"
701px / 96ppi = 7.3021"
Select the image, and it should show at the above size, displayed in inches, and Illustrator has done its job properly. Illustrator is scaling the image to the physical size calculated by the pixel dimensions divided by the PPI. This is what it is designed to do.
Don't confuse Illustrator's "pixels" with the true pixel dimensions of the image. Illustrator technically has no pixels, since it's a vector image editor, not a raster image editor. But it uses 72ppi as its default for displaying images.
You will notice that if you set the General Units to pixels in the preferences, it will now display the size as 675px x 525.75px. This is absolutely normal. You can ignore this, since it's not the true size of the image file. Let me explain . . .
What Illustrator is doing here is calculating the pixels according to Illustrator's default 72ppi. Essentially, 9.375" x 72ppi = 675px to make the image appear at 9.375", which is the physical size indicated by the ppi of your original image (which is 96ppi).
You can check that the image is still the same size as it was originally in pixels. Select the image, and in the tool options along the top, click on the "image" link, hit the tiny menu icon and choose Link File Info, select the Camera Data tab, and scroll down until you see the pixel dimensions of the image. You will see that it's still 900px x 701px.
Here's an example: Note the size shown in pixels in Illustrator is not the actual size of the image
If you don't want Illustrator to resize the image, then in Photoshop, set the PPI to 72 (making sure you don't resample the image), then Illustrator won't resize it when you place the image. When you set the ppi to 72, you are basically telling Illustrator to ignore the PPI.