The legislation is seen by the government as crucial to overcoming existing legal barriers to the scheme, under which the government wants to start sending asylum seekers arriving illegally in Britain to Rwanda to have their claims processed.
However, both chambers of Britain's parliament must agree the wording of the bill before it can become law and on Wednesday the upper chamber voted in favour of making changes to the version approved by the lower house.
Those changes are now expected to be rejected again by the lower house in the latest leg of a protracted parliamentary battle that underscores the divisive nature of the scheme.
The legislation is still ultimately expected to be approved by both houses, albeit later than Sunak had hoped. By informal convention the unelected upper chamber is expected to drop its proposals if the elected lower chamber keeps rejecting them.
Facing an election later this year that he is widely expected to lose, Sunak is hoping for a popularity boost by getting flights to Rwanda started as soon as possible to try to meet his pledge to "stop the boats" crossing to Britain.
But with charities and human rights groups saying they are gearing up to bring legal challenges, Sunak might not meet his deadline of getting the flights under way by the end of Spring.
(Reporting by Elizabeth Piper and William James; Editing by Alexandra Hudson)
