Bicep strain? tear? Treatment?

A month ago, in the process of doing an inelegant curl with a heavy bar, I experienced a slight jarring sensation in my left bicep and stopped exercising immediately. Ever since, I have felt a mild, occasionally radiating pain in my left bicep, though it sometimes disappears entirely from hour to hour, even from day to day. Range of motion is unaffected and it is not sensitive to the touch.
I apply ice twice a day and I’ve stopped doing curls, but the condition remains unchanged. (Curiously, when I do pushups, it seems to banish for the pain for some time afterwards!) Am I describing a bicep strain? A tear? An injured tendon? And is there some value in doing pushups as a form of physical therapy? Are there other helpful exercises to speed healing?