The situation has become so desperate that Glasgow Royal Infirmary has only one donor on its books - and his sperm can be used for a maximum of just 10 pregnancies.
"We have a chronic shortage," said Dr Richard Fleming. "We have tried advertising in GPs' clinics and student unions but it hasn't been successful. The situation has become critical."
The situation is altogether healthier at the Cryos sperm bank in Aarhus, Denmark. It has 200 suitable donors, though they are paid a mere pounds 25 if their sperm is of good enough quality.
"I don't know why we have so many more sperm donors in Denmark but we obviously do," said a modest spokesman for the company yesterday.
In the past, the GRI relied on students, but a number of reasons have led to a fall in the number of donors.
Firstly, the Human Fertilisation and Embryology Authority (HFEA) has said that payment of sperm donors must end (at present donors receive pounds 15 plus certain expenses). Secondly, as well as much more stringent screening, there are fears that one day, children born after artificial insemination may have the right to know the identity of their genetic fathers, and prospective donors are warned about this.
Dr Fleming has put in a formal application to the HFEA asking to import sperm in bulk - at present forbidden under the authority's rules. A spokesman for the HFEA said yesterday that it had received the application and was taking legal advice.
But Dr Fleming argues that it is far more practical to go to Denmark than rely on Britain's native population to be altruistic.
He said: "An English sperm bank did do a lot of advertising in Bristol - spending pounds 30,000 in total - and only one man came forward."