The easing of Covid restrictions and reopening of various sectors meant demand for workers rose at its fastest rate in May for more than 23 years, according to a KPMG survey.
But the number of staff available to fill those jobs declined at the quickest rate since 2017.
KPMG called on the government and firms to address skills gaps.
Workers were especially needed in IT and computing, which has been a long-term trend, as well as hospitality, the survey found.
Hospitality venues are struggling to fill thousands of job vacancies with waiting staff and chefs in particular demand as Covid-19 restrictions continue to ease, industry body UK Hospitality said last week.
Many hospitality firms have struggled during the pandemic despite extensive government support, including rates holidays and the furlough scheme, and staff have lost their jobs or seen their hours reduced.
The retail industry saw the least demand for workers.
This monthly survey is based on responses from four hundred recruitment and employment agencies and is yet more evidence of the economic recovery gathering pace.
But there's also a shortage of workers. Hospitality and events firms have been struggling to get the staff they need to cope with a post-lockdown surge in demand.
Some are warning of a staffing crisis this summer, partly caused by gaps left by EU workers who have returned home because of Brexit and the pandemic.
The survey's authors say skills and labour shortages, that already existed before the pandemic, are now the most pressing issue in the jobs market.
This means many employers will need to look at the wages and benefits they offer at a time when they're still struggling to financially recover from the last 14 months.
The jobs market "seems to be firing on all cylinders" said KPMG partner Claire Warnes.
"But the deterioration in staff supply intensified this month... This is a worrying trend," she added.
She called for government and firms to "urgently address the skills gap".
Kate Shoesmith, deputy chief executive of the Recruitment & Employment Confederation (REC), said: "With demand spiking, the skills and labour shortages that already existed in the UK have come into sharper focus - and Covid has only made them worse.
"This is the most pressing issue in the jobs market right now, and has the potential to slow down the recovery."
A government spokesperson said: "Our multi-billion pound plan for jobs, including the Kickstart scheme, is supporting employers across the country to create jobs and help jobseekers get the skills and experience needed to develop their careers and fulfil roles for years to come.
"Additionally, we have increased the apprenticeship hiring incentive to £3,000 per new apprentice hire and our lifetime skills guarantee - worth the equivalent of around £3,400 per person - ensures all adults can gain new skills and qualifications."