Full details will be announced in the coming days, a Number 10 spokesman said.
A Downing Street spokesman said today: "As the PM said in the House of Commons, he has been thinking about this a great deal".
"Every death in this crisis is a tragedy, and sadly some NHS support staff and social care workers have made the ultimate sacrifice in the pursuit of saving the lives of others", the Indian-origin Cabinet minister said.
But he added: "On the other hand we must look at the realities - this is a great national service, it's a national institution, it needs funding and those contributions actually help us to raise about £900 million, and it's very hard in the current circumstances to find alternative sources".
"NHS and care workers from overseas who are granted visas are doing this already by the fantastic contribution which they make".
The IHS is a fee that foreign nationals have to pay to use the NHS.
The £400 surcharge remains in place for other categories of visa applicants and will increase to £624 in October, as planned.
It also includes independent health workers and social care workers.
Senior Tories demanded change, with former party chairman Lord Patten calling it "appalling" and "monstrous".
"Personally, I believe that politically courageous and sensible politicians have the ability to revisit positions and to put something right if it has gone wrong".
Public Administration and Constitutional Affairs Committee chairman William Wragg called for an immediate change in policy, adding "now is the time for a generosity of spirit towards those who have done so much good".
Mr Johnson had said yesterday during PMQs that he understood the opposition to charging foreign NHS workers but said the United Kingdom must "face the realities" of the funding that the fee provides the health service. Scrapping the NHS charge for health and care workers from overseas is the right thing to do. "These workers have our backs, we will always have theirs".
"The scrapping of the surcharge must come into effect immediately to show our gratitude for the effort and dedication our overseas workers give to the NHS and to their patients".
"I welcome the news on their behalf, but it is a shame it took this pandemic for the Government to see sense". The Home Office said a dedicated team will work directly with NHS Trusts and employers to identify family members to whom this extension may apply and arrange for the issue of indefinite leave to remain to them.
The extension follows criticism of the government's scheme by the Opposition Labour Party and workers' union after it emerged that it would not cover lower-paid workers such as cleaners and porters working within the healthcare sector.