UK ministers said they are still considering other options, including mine-hunting drones, but have not committed naval vessels to the operation.
According to The Telegraph, France, Germany and South Korea have also signalled reluctance to take part, as concerns grow internationally about the risks of further escalation in the conflict.
The report said oil prices have already jumped to about $106 a barrel, reflecting fears of disruption to shipping through the waterway, which carries roughly a fifth of global oil supplies.