If you are in the US, you have very little to be concerned, unless you have something negative written against you in your HR file. Even in some cases, past or soon-to-be-past employers do not bring them to the table, with the fear of being sued.
Provided your company A is a structured organization with HR department and a legal department and such. In that case even if they call your boss directly, he had better not talk to those guys but refer them to HR for the proper answers. And HR will only verify your employment dates and your title or titles if you had more than one during your tenure at company A.
So, by all means, give company B the number to HR department of company A and tell them that the procedure to contact your supervisor is to ask HR to connect them. In no company I worked in US would do that. And since you are in the final leg of your employment with company, does it really matter if they know who your next employer will be ? unless of course you are going to work for a direct competitor and have signed non-compete clauses in your contract, in which case I advise you to lawyer up, quick.