Dr Malvern Tipping, owner of the cliff railway, said staff remained "anxious" about the impending 'Freedom Day', and that he expected the measure to remain in place for all passengers apart from those legally exempt.
It comes after transport operators across the county issued differing statements on what they would be enforcing.
Transport for Wales said passengers will be required to wear face masks on all of its trains travelling in England and Wales.
West Midlands Railway said it would follow guidelines set by the Rail Delivery Group (RDG) and only ask people to wear a face covering if an indoor setting is "busy".
Meanwhile, Arriva Bus suggested it would follow guidance from the Confederation of Passenger Transport (CPT), which said passengers would not be forced to wear face coverings in Shropshire and Telford & Wrekin.
Dr Tipping said: "It has already been reported to me by the general manager that our staff remain anxious about any relaxation of face mask wearing at the cliff railway.
"It has been reported to me that most of our passengers seem to prefer that there be face mask wearing measures in place.
"The view of the Bridgnorth Cliff Railway directors is that we should follow the lead of Transport for London in making face mask wearing a condition of carriage for all passengers apart from those who are legally exempt.
"Before making a final decision, the directors will consult with the general manager and members of staff. Their views and safety are paramount.
"So I expect that face mask wearing will become a condition of travel except for the legally exempt.
"I think that model of Sadiq Khan, mayor of Greater London, in making mask wearing a condition of travel on public transport, apart from those who have a legal exemption, is very much the way forward until such a time that the pandemic recedes."