We have On-Premises Team Foundation server 2013. We are planing to use the Visual Studio Team services.
Can we use the on-premises TFS user account for Visual Studio Team services?
On premise TFS accounts will be based on local active directory while VSTS you can have users with microsoft id or users in Azure Active directory. So if you migrate the users to azure active directory or sync local AD to Azure AD using Azure AD connect you should be able to achieve the functionality you are looking for.