Hunt has argued that doctors and nurses should be temporarily exempted from the cap until the Government's planned 25 per cent increase in training places starts to have an effect.
Health chiefs have called the cap "absolutely barmy" after official figures revealed that 1518 bids for visas by foreign doctors were turned down in the four months to March.
The move represents a significant softening of the Prime Minister's tough line on immigration and comes ahead of the 70th anniversary of the NHS next month.
May is likely to face pressure from other members of her Cabinet to exempt other foreign workers in highly skilled professions from the cap.
Between December and March, more than 6000 visa applications were turned down from workers including doctors and scientists.
The Prime Minister has faced a series of challenges over her approach to immigration, with ministers urging her to drop her target to reduce the number of people coming to the UK to "tens of thousands". She has also faced mounting criticism of the "hostile environment" for illegal immigrants in the wake of the Windrush scandal.
Javid confirmed that the Government would take a "fresh look" at the overall cap on "tier 2" visas for highly-skilled workers. He also repeatedly refused to endorse the Prime Minister's target for net migration.
He said that "a number of doctors that are qualified, that our NHS needs, are being turned away".
The Home Secretary also echoed Ruth Davidson, the Scottish Conservative leader, in suggesting that students should be removed from the net migration target.
On the Windrush immigration scandal, Javid said that 32 of the 63 people who were wrongly deported under the policy were convicted criminals who would not be allowed to return to the UK.
The remaining 31 were being located and would be compensated, he said, as he confirmed that he would review "aspects" of the May's "hostile environment" strategy.