Yes, you can use the Export, Import Process

Have you tried exporting the webpart  from the source area  then import to
target area?

Can you export it from within Sharepoint? I don't see any such option

Edit, page, select the down arrow on the web part and hit Export

I can export custom webparts, however I can't export webparts which are just standard webparts.

Edit page >> Modify page >> Modify Shared webparts

Then click on the downwards triangle and select export