P.S. In another question you suggest that you are talking about part of a machine. If it is a relatively large, heavy item, and it needs to be shipped in a big, strong container, in U.S. English it would be common to call the container a "crate" and talk about "crating" the item.

To "pack" is to put things into a container that already exists. You can pack a suitcase. You can pack the trunk of a car. We bought a souvenir of our trip, packed it into our luggage, and brought it home. To "package" means you are also creating the container itself. To send a gift, you wrap it in bubble wrap, put it in a cardboard box, tape the box shut, address it, and so forth. You are creating the package and thus you are "packaging" the gift. We bought a souvenir of our trip, packaged it, and shipped it to our grandkids.

Hey, I would say that you can, 'package an item,' or, 'send and package,' whereas you would, 'pack your bag (for travelling).'

Package is always the verb I'd use in regards to shipping or sending an item. Pack is a verb used more toward "packing a lunch" "packing your bag for a trip" "packing up your tent equipment" etc. EDIT: To answer your above comment, I also agree that you can say "pack the item with..." because it's the same idea as above, as if you're packing a bag, you're packing a bag/box with an item in preparation to ship the package.

I think the right way is pack.