Two sources said that Ms Coffey, a long-standing ally of Ms Truss, was a favoured candidate for the position as NHS backlogs are considered a critical issue ahead of the next general election, expected in 2024.
The move would mark a promotion for Ms Coffey, from her current role as secretary of state at the Department for Work and Pensions.
Whitehall insiders said that she had been seen meeting with the current permanent secretary, the most senior civil servant, in the health department. Though one official said that if Ms Coffey were to take on a Cabinet Office role, such a briefing would also be appropriate.
The top jobs at what are traditionally termed the three great offices of state – chancellor, foreign secretary and home secretary – are expected to change hands.
Priti Patel has told staff that she expects to leave her role at the Home Office.
“She’s been clear to staff that she doesn’t expect to remain, that’s been made very clear to staff,” a home office source said.
Kwasi Kwarteng, another close ally of Truss, is set to be her neighbour in No 11 Downing Street as chancellor, if she wins the contest.
Mr Kwarteng told senior business bosses that he was likely to end up with the top job in the Treasury in a recent meeting, according to sources familiar with the discussions.
Mr Kwarteng has also been joined by Jacob Rees Mogg in sensitive meetings about energy security, fuelling speculation that he would be given the business secretary brief.
James Cleverly is widely tipped to become foreign secretary under a Truss administration. Until taking on the role of Education secretary this summer, Mr Cleverly had held a range of foreign policy-related roles, and worked closely with Liz Truss during her time as secretary of state in the foreign office.
“Given the current challenges, someone who has a bit of a grasp of geopolitics – as Cleverly does – would be helpful,” a foreign office source said.
Ben Wallace, who lent his backing to the Truss campaign in July, is also set to stay in his current sensitive role of Defence Secretary, according to MoD insiders.
Anne-Marie Trevelyan is also expected to stay in her role as trade secretary.
The Conservative membership vote to determine the next prime minister will close on Friday following last-minute messages from supporters calling party members to cast their ballots on social media.
The victor will be announced on Monday, shortly after midday. They then face a journey north to the Queen’s estate in Balmoral, Scotland, in order to be formally appointed to lead Her Majesty’s Government.
The Truss campaign declined to comment. A spokesperson for Ms Patel did not respond to a request for comment.