run a fever
To have an abnormally high body temperature (a fever), which is indicative of or caused by illness; to have a fever. After my toddler was lethargic all day, I begin to worry that she was running a fever.
run a temperature
To have an abnormally high body temperature (a fever), which is indicative of or caused by illness. Dan: "How's Pete feeling?" Marshall: "Well, he ran a temperature last night, but he seemed a lot better this morning after some rest." I think I've started running a temperature. Maybe I should go lie down.