In April, family proceedings became open to the media for the first time, but tight restrictions meant that only limited information could be reported.
But Straw has said he will now ask the Family Procedure Rules Committee to consider whether court rules can be made to allow journalists to report the "substance" of children's cases.
Under the reforms courts will still be able to use existing statutory powers to protect the identity of those involved.
But the government said it will be working with the media to explore ways in which the privacy of families can be protected.
Jack Straw, Secretary of State for Justice, said: "Family courts play a very important role in our society, and the decisions they make need to be transparent so that the public who use them have confidence in them.
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