The new vaccine would contain four different viruses
But it will be at least two years before GlaxoSmithKline submits its MMRV vaccine for approval in Britain.
One theory is that subjecting the body's immune system to three types of virus in one go is simply too much.
This claim has been rejected by the medical establishment, who are adamant that the vaccine is safe, and that a drop in immunisation rates risks a re-emergence of disease.
However, there is increasing demand for single dose vaccines for measles, mumps and rubella in the UK.
The new vaccine, which would include a dose of the varicella or chickenpox vaccine, is sure to provoke more controversy.
Cheaper alternative
A GlaxoSmithKline spokesman said the vaccine would work out cheaper than MMR because it cuts down on the number of injections a child has to undergo.
Combined vaccines are used extensively in countries where there has been less controversy over the MMR jab.
Currently, licensed chickenpox jabs are not available in the UK although they can be obtained through some private companies.
Although the disease is not seen as dangerous, latest figures show it accounts for about 25 deaths annually in England and Wales.
Topically, this is more than from measles, mumps, whooping cough, and Hib meningitis combined.
A Department of Health spokeswoman said: "We are not in any way considering adding any vaccine to MMR at this point.
"If and when such a vaccine is licensed we will consider all the evidence on its safety and efficacy in combating chickenpox and shingles."
Chickenpox can be fatal, especially in people with suppressed immune systems, and deaths from the disease in adults have increased over the past 30 years, despite the availability of a vaccine against it.
Once someone has had it they become immune to the disease, but the virus does remain dormant in nerve tissues and may reappear to cause shingles (herpes zoster) in later life.