If the short bursts of tuition succeed in helping pupils in pilot areas, it could be rolled out across England.
The NUT welcomed the plans, saying one-to-one tuition should not just be restricted to those who can afford it.
The General Secretary of the National Union of Teachers, Steve Sinnott, told the BBC: "If a youngster is struggling at school and their parents have the money to be able to assist that youngster, what they do is they go and they get them a private tutor.
"We believe that access to that one-to-one tuition shouldn't be restricted simply to the children of those whose parents can afford it."
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