I am assuming that the links are within formulas in the destination workbook. If so, this has been a problem for a very long time.
The workaround is instead of incorporating the link in a formula, create direct links from the source workbook to a separate worksheet in the destination workbook and then in the formulas reference the linked worksheet which is now in the same workbook. This way, the links to the closed workbook work.
The linked sheet can be hidden.
Just some additional information with links. If you want to move or rename workbooks containing links then open all of the workbooks and keep them all open and use "Save As" to either rename or save the workbooks to a new location. Using this method, Excel automatically renames the workbooks in all of the associated links.