Under the deal, announced by Education Secretary Michael Gove, bank staff will sit on governing boards, provide financial advice to schools and offer work experience opportunities to pupils.
The Barclays Money Skills education programme will be offered to free schools and academies, while teachers and other school staff will have access to online business skills training materials.
Free schools and academies will also be offered free banking accounts with the bank.
Christine Blower, NUT general secretary, said: "Opening schools up to the market place is simply wrong. Children and young people should not be influenced at an impressionable age by whichever large company manages to gain a foothold in their school.
"Any successful business’s involvement in a school will surely be decided on what returns they can reap for themselves."
Gove wants to extend private sector involvement in running free schools and academies further.
He said: "Since the 16th century, when the first City livery companies opened schools, we have seen huge benefits when schools and businesses work together. I want a new generation of businesses to do the same in our changing schools system."
Barclays retail and business banking chief executive Antony Jenkins also defended this latest link up.
He said: "Barclays is supporting free schools and academies because we want to boost financial skills for young people. By providing financial awareness training and valuable work experience we can help young people to contribute to and share in future prosperity."
