The agency said that a step to over-the-counter antibiotics could make the problem worse, for doctors are running out of medications for potentially deadly infections.
And rising resistance levels mean that some bugs are treatable only with antibiotics previously used as a 'last defence', the agency added.
The number of potentially fatal E.coli blood poisoning cases, which are resistant to a 'last' antibiotic have tripled in the last six years, and now make up 12 per cent of all cases.
In these cases, doctors are left able to use only less effective antibiotics, which can be toxic, and raises the prospect that a strain could become entirely resistant to antibiotics.
Dr David Livermore, the agency's top scientist, warned that the problem posed a major public health threat.
He suggested that patients should not request antibiotics from their GPs when they are simply suffering from a cough or a cold, for which the drugs are useless.
"Most common cough and colds are caused by viruses and therefore patients should not be asking their doctor for an antibiotic," the Telegraph quoted him, as saying.
"The doctor should also know that they are not necessary and the patient should know that they are not appropriate," he added.
Livermore also said that there was a concern that providing an antibiotic over the counter could increase resistance.
He also called for action to increase the amount of research into new antibiotics.