When we moved into our current flat, because I am self employed the landlady needed my mum to act as guarantor. We got the forms through for the credit check, and it asked for NI numbers. My mum point blank refused to give it, I spoke to the letting agency, and thay said 'no problem'. We all passed the check.
The NI number may be asked for, but a check can be done without it.
Vicky