While I initially laughed at the idea, apparently there might be:
Hotel Islazul, Guantánamo city may be able to arrange tours to the
American military for around $40 though this service is not always
It does seem to be a long shot though. It's a US Military base, which generally means off limits unless you're enlisted, family, or have an invitation.
Certainly there is evidence of media tours, if you're a journalist or in a related field you may have a better chance. But it sounds like Hotel Islazul might be your best shot - perhaps contact them when you arrive. According to CubaWeb, their contact details would be:
Hotel Islazul Guantánamo
** Address: Ahogados y 13 Norte, Guantánamo.
Phones. (5321) 38 1015 / 38 2535 / 38 4555