The minister for the Lord Chancellor's office, Geoff Hoon, was tempted to go along with the deal, because it would have helped dozens of British families in "tug-of-love" cases in the US.
The sticking-point was a demand that, in return for making it easier for Britons to seek custody for their children in US courts, US citizens should be entitled to easier access to justice in the British courts, including legal aid.
The Government is limiting its availability but the US request raised the prospect of Hollywood wives with British connections using legal aid to sort out their arguments over divorce in this country.
"He thought long and hard about it before saying no, but in the end he had to turn it down," said a Whitehall source.
The decision to turn down the US offer will disappoint British families engaged in drawn-out custody battles for their children through the American courts.
They face daunting difficulties in going ahead with legal action in the US courts in the federal system, where state courts have responsibility for family law.
The US government, in private talks with the Lord Chancellor's office, offered to simplify the system by allowing Britons to seek justice through federal courts instead of negotiating through a maze of lower courts, which can prove expensive.
The cases have included that of Sheehan and Conar Sidwell, who were snatched by their father and taken to America via London and Germany.
Their mother, Kim, tracked them to Orlando and brought them home after a legal battle but was left with a pounds 6,000 bill.
Scrapping routine checks on passports of UK citizens leaving Britain has been blamed for a 58 per cent increase in parental abductions of children from this country to live abroad since 1995.Reuse content