My experience in life is that nearly everything is negotiable.

First, every large company has someone who handles Human Resources, and also someone who does payroll, and usually they are the ones who verify employment for loan applications, etc. not your first line supervisors.  That is the number you should have them call.

Second, if for some weird reason this is not an option, there are other ways.  They don't really care whether you are employed or not, they just want to verify that you are who you claim to be.  They verify retired people all the time and they have no employer to call.  I am self employed and had no problems.  So if you have any kind of internet presence that will usually work.  Google your name, does anything come up?  There are other ways too, just call directly and they will work with you.

There many ways to verify just call and they will be happy to work with you to find a way

Never ever ever provide your work information. You open yourself up to nasty consequences. You can verify very easily with past providers without giving out employment information.