Not all cities welcome additional immigration, but perhaps those that do could sponsor visa holders. The visa could be temporary and renewable, with a path to permanent residency and eventually citizenship. Visa holders would be free to bring their immediate family members with them...
A policy that allows a greater number of law-abiding immigrants into the American cities that want them most could do more for global welfare than other policies related to trade and aid. An effective policy of this sort would be a win-win—a way for struggling American cities to stabilize their populations and a way for immigrant families to live, work, and study in the United States.
The point is easily transferable to the UK, as well; though the reasoning is more political than economic.
London is far, far more pro-immigration than the rest of the UK. In a 2011 Ipsos MORI poll (pdf), 61 per cent of Londoners thought immigration is "a very big or fairly big problem", compared to a UK average of 75 per cent and a peak of 88 per cent in the West Midlands. Similarly, although 15 per cent of Britons opposed a cap on the number of non-EU workers, 23 per cent of Londoners did.
Obviously those figures still don't reveal a populace willing to welcome all-comers with open arms. But they do hint that a national anti-immigrant agenda could be tempered if London were given the power to control its own borders.
Of course, such a move wouldn't do wonders for relations between the capital and the rest of the country. Given the fact that the success which has already accrued to the capital from having an elected mayor has lead to calls to scrap the position to aid a "levelling down", it's hard to imagine the bad blood that could result from an immigration-driven economic boost.