The Society and College of Radiographers says there are an increasing number of reports of patients trying to make recordings.
But it warns sonographers could be distracted, and miss vital signs of problems in the developing baby.
It is down to individual hospitals to decide if recordings should be allowed.
The college, which says that developments in technology mean the ways in which such recordings could be made are bound to increase, has now issued guidance to its members on how to deal with patients' requests to film scans.
It says that, as well as the risk of diverting the sonographer's attention at the point when they most need to concentrate, there could be legal complications if an abnormality is recorded by the scan, but not reported or acted upon.
Many hospital trusts allow parents-to-be to take away stills from the scan, usually for a charge.