Two-thirds of such jobs could go to women by 2020, says the UK Commission for Employment and Skills.
UKCES warns that men urgently need to improve their levels of expertise if they want to compete.
TUC general secretary Frances O’Grady said she feared better skills might not lead to better pay and jobs for women.
“The increased disparity between men’s and women’s skill levels is concerning for both sexes.”
The researchers drew on 10 years of quarterly data on skills levels from the UK Labour Force Survey, which gathers data on 54,000 people of working age.
They were particularly interested in the numbers with “higher-level qualifications” or above – for example Higher National Certificates or Certificates of Higher Education, which can count towards university degrees.
The overall proportion of the adult population with higher academic or vocational qualifications rose from just over a quarter (25.7%) in 2002, to well over a third (37.1%) in 2012.
By 2012, 38.3% of women had such a qualification, compared with 35.7% of men. If trends over the past 10 years continue, researchers predict that by 2020 almost half of women (49%) will have this level of qualification, compared with 44% of men.
The data also suggests the proportion of women with only a low level of qualifications – for example, without five good GCSEs in England. Wales and Northern Ireland or National 5 certificates in Scotland – is likely drop to 15.4%, compared with 20% of men.
“Tackling inequality, in skills, qualifications and pay, and for both sexes, is essential if we are to have a prosperous and stable future,” said Ms O’Grady.
“Men are finding it harder to get skilled jobs, while for many women their higher qualifications are not leading to better pay and jobs.”
The report also warns that although the skills level in the UK is increasing, other nations may improve faster.
The researchers are particularly concerned that although the numbers with high-level qualifications is increasing, there is still a “long tail” of people with low skills.
UKCES chief executive Michael Davis said addressing low skills was vital to sustaining growth over the long term.
“These projections are a warning shot for our future selves.
“They highlight the importance of employers continuing to develop the skills of all their employees to ensure that businesses compete successfully and the economy continues to grow.”
The researchers say having qualifications is a “key measure” of skill levels though they accept it is “imperfect”.
They note that many individuals possess skills highly valued by employers but hold no formal qualifications, while employers may be sceptical of the value of some qualifications.
They add that a general improvement in qualification levels may be of limited benefit if employees do not gain “economically valuable skills which employers demand”.