Free movement
The right of citizens of EU member states to move freely around the European Union for employment purposes is one of the four fundamental freedoms guaranteed by the EU. Both the Home Office (the UK governmental department responsible for work visas, among other things) and the FA must comply with the free movement protections guaranteed by the European Union.
If the UK left the EU, then UK footballers would be unlikely to continue moving freely to clubs in EU member states. Similarly, EU footballers would be unlikely to continue moving freely to UK clubs.
Mill's & Reeve discusses the implications in a full length article - read it here.
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