Since September, applicants for Early Years Educator (EYE) Level 3 courses are required to have maths and English GCSEs at grade A to C upon completion of the training if they are to count towards a setting's staff-to-child ratios.
The recent National Day Nursery Association's (NDNA) workforce survey found that early years settings were facing a "catastrophic recruitment crisis" because of the requirements.
Announcing the consultation, Dinenage said the government wanted to get the views of early years providers and practitioners on how to ensure the workforce has sufficient skills to deliver the 30 hours of free childcare, when it is introduced for three- and four-year-olds in September 2017.
The consultation will inform the government's early years workforce strategy, which is expected to be launched in late autumn.
Dinenage said: "Making sure our children learn, develop and flourish at this critical time in their lives is vital - we want to recruit and retain the very best staff for this, that's why we are looking at the skills needed.
"The findings will help inform my upcoming workforce strategy which, along with our record investment in the sector, will support nurseries, pre-schools and other early years providers to offer high quality, free childcare for thousands more families across the country."
Last month, CYP Now reported that some training providers were refusing to take on learners who did not have the required GCSE grades, essentially making them entry requriements.
Neil Leitch, chief executive of the Pre-school Learning Alliance, said it is positive that the government is reconsidering its policy.
"To say that this is causing a recruitment crisis is not an exaggeration, and clearly, blocking talented, experienced and passionate practitioners from entering the sector at a time when we are looking to expand the free entitlement offer is both illogical and impractical.
"We urge as many practitioners as possible to respond to the consultation, and hope that the government will listen to the views of the sector on this issue and look to reverse this policy as quickly as possible."
Liz Bayram, chief executive of the Professional Association for Childcare and Early Years (Pacey), described the consultation as an "important first step".
"The consultation must be part of a coherent workforce development strategy or we risk losing great practitioners, who feel undervalued just when the sector needs them to help it meet the expected demand for the enhanced 30 hours entitlement," she said.
Julie Hyde, associate director of CACHE, the organisation leading the Save Our Early Years Campaign, called for functional skills instead of maths and English to be used to assess apprentices' abilities.
"This is very encouraging and we hope it paves the way to remove the barriers to recruitment in the early years," she said.
Stella Ziolkowski, director of quality and workforce development at NDNA, said: "We welcome this much-awaited open consultation on literacy and numeracy skills ahead of the forthcoming 30 hours childcare reform and urge everyone including practitioners in early years to get involved and have their say.
"NDNA has long campaigned for a review of these requirements as part of wider strategies to make sure the nursery sector has a well-qualified, professional and highly-trained workforce."
The consultation will run until Monday 28 November.