According to ARLA Propertymark's latest figures, the average number of prospective tenants registered in each branch of an estate agency increased to 107 in August from 102 in July.
This record is broken by the previous month's record.
Data also show that rent increases for tenants increased significantly in August's second month. 79% of agents reported seeing landlords increase rents, compared to 71% one month earlier.
This is the largest rate increase ever recorded, surpassing the May record of 68% for the second consecutive month.
Scotland was the region with the most properties managed per branch by a letting agent, at 251. London had the lowest rental stock with an average of 126 properties per branch.
Nathan Emerson, Propertymark's CEO, said: "The constant increase in tenant demand coupled with the declining rental stock means that we are seeing a worrying rise in rent prices for a second month. Private rented housing provides homes for around five million households in the UK. It is important that landlords are recognized for their service. The increasing pressure on landlords means that more landlords must be encouraged to enter the sector.
"We look forward meeting the Secretary of State and his staff over the next months to ensure that keeping the rent flowing remains high on the Department’s agenda moving forward. We also have a close watch on the sector for the remainder of the year because of growing concerns about house buyers leaving the buying and selling markets. This will put additional pressure on private rental sector."