Link and Embed
There are two ways to save design images without missing any data in Illustrator presented as below.
Check the link box when placing a file.
Uncheck the link box when placing a file and select.
Use Embed function when link box is checked
Select Embed on the top menu after selecting the image.

Uncheck the link box when placing a file and select.

Select Embed on the top menu after selecting the image.