Former shadow public health minister Diane Abbott dubbed new immigration legislation "a nasty little bill" and warned it would demonise British people of colour.
Some plans were already covered by British law, such as preventing illegal immigrants from getting NHS care, she told Daybreak.
"If the Government wanted to help with that it could help hospitals collect the money," she added.
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Four men are being interrogated in Spain's National Court over the vehicle attacks that killed 15 people and wounded more than 120 others.
In the position paper the Government sets out plans to ensure that cross-border legal disputes are dealt with in a "fair way" post-Brexit.