But now she is poised to abandon the idea under pressure from pro-EU ministers including Philip Hammond and Amber Rudd, it was claimed.
The policy would have made companies complete an “economic needs test” when looking for new workers, to check whether they could fill the vacancy with someone from the local area.
Only once they were sure there was no suitable Brit would they be able to hire a replacement from an EU country or beyond.
The plan – similar to the system used in the US – came to light as part of a leaked document detailing a possible new immigration framework for Britain after we leave the EU.
The Government has repeatedly said they want to train up British workers so they are not leapfrogged by more highly skilled foreigners.
But after Mrs May met with senior ministers to discuss the future of immigration, the economic needs test was scrapped, according to The Times.
A source told the newspaper: “Rudd and Hammond made sure that got taken out.
“It’s been through about eight different versions and each one is slightly less hardline than before.”
The news could enrage Brexiteers who are keen to lower sky-high migration figures after we quit the EU and no longer have to accept the free movement of European workers.
The Chancellor and Home Secretary say that cutting immigration too fast would hit business and damage the economy.