Health and Social Care Secretary Matt Hancock announced over the Bank Holiday weekend he plans to run for prime minister.
He is joined by nine other Conservative MPs, which includes former health secretary Jeremy Hunt. More names are expected.
On Friday Prime Minister Theresa May announced she would be quitting on 7 June, with a new leader possibly in place by the end of July.
Niall Dickson, chief executive of the NHS Confederation, which represents organisations across the healthcare sector, praised May for delivering a long-term funding settlement for the NHS during her time in office.
However, he added: ‘Yet she entered Number 10 pledging to fight injustice and failed to tackle one of the country’s biggest injustices – thousands of the most vulnerable in our communities are being left without the care and support they need, and millions of family carers are exhausted and at the end of their tether.
‘This is the greatest social challenge of our time – the next Prime Minister cannot duck this tragedy in our midst and should be judged on what they do to tackle the immediate crisis and to develop a long term solution.’