Greg Sargent discovers a contemporaneous account of Mitt Romney swearing off the idea of putting a Muslim in his cabinet from before this issue became controversial and Romney started denying he'd ever said any such thing:
So when Republican presidential candidate Mitt Romney recently addressed a group of a prominent local conservatives at a Las Vegas fundraiser, George lobbed the first question: “If you are elected President,” he asked, “will you include any Muslim members in your cabinet?”
In the seconds before former Massachusetts Governor Romney responded, you could have heard a pin drop.
His (admittedly, very smooth) answer in a nutshell? “Not likely.”
Now unfortunately we don't know what the exact terms of his "admittedly, very smooth" answer were from this account, but it's clear if you read on that people who think Muslim candidates should be considered weren't happy with his response.
And I'll note once again that for Romney this isn't a merely hypothetical consideration. I'm not sure that there are any Muslim Democrats who stand out as obviously choices for cabinet jobs (which isn't to say you couldn't find someone qualified, but there's not an obvious choice), but it's not at all the same on the Republican side. UN Ambassador Zalmay Khalilzad is a Muslim and he's serving the incumbent administration in a position that's traditionally been a waystation for people on their way up to cabinet appointments or agency chief jobs. It'd be really odd for a Republican administration to not consider him, and repugnant to not consider him because he's a Muslim and there aren't enough Muslims in the country to make them deserve a cabinet spot.
Defense Department photo by Spc. Michael Pfaff