The bankers, known as the “NatWest 3”, were jailed in 2008 for their role in a fraud case related to the collapse of energy trader Enron.
The three had lost a long legal fight against extradition to the United States.
Now British MPs are seeking changes to the extradition laws and will debate the issue on Monday evening.
Campaigners in Britain argue that the rules are lopsided and have taken up the ongoing case of accused British computer hacker Gary McKinnon, 45.
He was arrested in 2002 after allegedly hacking into US security systems including the Pentagon and NASA and is trying to avoid being sent to the United States.
On Monday McKinnon’s mother wrote a letter published in the Daily Mail pleading with legislators to change the law to save her son from extradition and subsequent imprisonment.
Critics in Britain say it is easier to extradite British citizens to the United States than vice versa, a claim rejected by US Ambassador Louis Susman.
“We believe that the existing US-UK Extradition Treaty works, is fair and balanced, promotes justice in both countries, and does not need to be changed,” Susman said, writing in the Daily Telegraph.
Susman said the differences were purely semantic.
“In practice, the “reasonable suspicion” test is the same as the US’s “probable cause”.
They are the standards that police officers in our respective countries must meet to justify an arrest,” he said.
Parliament will debate the rules later on Monday and hold a vote which is not binding, but could increase pressure on the Conservative-led government to press the US authorities for changes to the 2003 treaty.
“We need a bit more common sense, a bit more discretion and a few more safeguards,” Conservative MP Dominic Raab told the BBC, adding: “We think in the interest of justice cases should be decided in open court with a judge.”