To improve Drupal performance, our client has suggested to change the field types from varchars to integers in some tables.
This isn't really the list for that sort of discussion. For the technical aspects of doing so, and whether it's a viable approach, you really want to ask on the development list, not the support list.
However, you'd really have to provide benchmarks that showed an amazing improvement in performance. IE., how does your client have any sort of clue that this is actually a good thing? Are they just waxing poetic about things they think they know something about, but don't?