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 tipped promotion for Ms Coffey comes as Tory MPs warn Ms Truss she must appoint MPs from all wings of the party, and not just a clique of loyal allies, if the party is to remain in power after the next election.
The top jobs at what are traditionally termed the three great offices of state – chancellor, foreign secretary and home secretary – are also expected to change hands if Ms Truss is unveiled as the new Tory leader on Monday.
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 Ms 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 Ms 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 expected to retain her role as trade secretary.
Tory MPs argue that one of Boris Johnson’s failings as prime minister was not to appoint a cabinet representing different sides of the party, and say Ms Truss must not make the same mistake.
“Because, it is as simple as this: if the government does not perform at an incredibly high level over the next few years, we are out of power,” one Tory MP said.
One former minister, a Rishi Sunak supporter, said a failure to take a “big tent” approach to forming a government could doom Ms Truss to swift rejection by her own MPs, a large majority of whom backed other candidates in the leadership race.
“If she appoints her team on the basis of loyalty rather than talent and doesn’t draw from all sides of the party, she is in danger of repeating Boris Johnson’s mistakes,” he said.
He added: “If you make it clear you’re not interested in what your MPs think, don’t be surprised if it comes back to bite you.”
The Conservative membership vote to determine the next prime minister has closed, 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.