Nationwide operations began this week to detain adult men and women, with more activity due to be carried out over the next 11 weeks leading up to a one-way flight to the east African nation.
Officials refused to say how many people had been held so far, but informed sources said there have been "dozens" of detentions across the UK, in cities including Glasgow, Liverpool, Birmingham and Bristol.
Enforcement action is said to have taken place in England, Wales, Scotland and Northern Ireland.
In a further development, the senior civil servants' union has submitted an application for a judicial review against the government's Rwanda plan.
The FDA has warned that its members could be in violation of the civil service code if they follow a minister's demands to ignore an urgent injunction from Strasbourg banning a deportation.
The Guardian disclosed on Sunday that detentions would begin on Monday. Several asylum seekers who turned up for routine Home Office appointments on Monday were detained and told that they will be sent to Rwanda.
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