Fears that the outbreak of the virus in west Africa could spread to other continents have grown in recent days, putting border staff across Europe and Asia on high alert.
Lucy Moreton, general secretary of the union that represents border forces in Britain, said that her members are "very concerned" abut the situation.
"They are phoning us up and asking 'what are we supposed to do, how do we spot this, how do we protect ourselves?', and we can't answer that for them just now," Moreton said.
"There is no health facility at the border, there is no containment facility and until extremely recently there has been no guidance issued to staff at all as to what they should do."
The warning comes a day after the British government's crisis response committee met on Wednesday to discuss its preparations to cope with any possible outbreak in Britain.
Foreign Secretary Philip Hammond has described it as a "very serious threat", amid concerns that a person carrying the virus could travel to Britain.
One possible case in Britain has tested negative.
Almost 700 people have been killed in west Africa by the virus since the first case was detected in February.