I did some searching, and found that Avis can provide car hire with driver service in the US. The program is called Avis Chauffeur Drive - WeDriveU.
As authorized drivers for the Avis rental vehicle, the chauffeurs can pick up and return the rental car for you, meet you at your hotel, business meeting or other location, and also provide mobile concierge services during your rental.
(opens in a new window) states that you may save up to 35% when using for
a full day as compared to using a car and driver service.
This option is available at participating cities , to Avis Preferred Members and certain Corporate account holders only.
Details on Avis Chauffeur Drive can be found on WeDriveU