The only thing I can suggest is you use the update email interface to do this by using DB rules to send update email commands between the 2 system.
When you create the new calls on the other system, record the older systems ID in a field on the new system. Let this trigger an email to update the old system with the ID on the new system in (for example)the Source ID field on the old system.
Then on the old system, this updating of Source ID could trigger an email back to the new system to update the calls there and use the option to send attachments.
I have not actually tried this but I think it might work.
Try creating the same attachment folder in the new system and copy all the attachments with the same name in that. Also, if u are using SD 5.X then in new system, do the same ftp settings as the current one.
I am agree with Ruth. But Ruth, I think veerat wants to migrate his existing attachments attached with the call..
Veerat: see hows the attachments are stored in your new system. Is there any field with each service call in new system pointng to the path of attached files? If so then create a text field in OVSD service call. Copy the attchment path details of each call in new field. Generate an excel report with all the details of the call along with that field and while uploading it into new system, point this new field with the attachment path field(if any) of the service call table of your new SD system.
Hope this helps.
And, I agree with Ruth..Its always better to assign points :-)