The reshuffle - the biggest of Mrs May's tenure - was prompted by the resignation of Damian Green after he admitted lying over pornography on his office computer.
According to The Sun newspaper, May wanted Hunt, who has been health secretary for five years, to move to become business secretary.
A desire to maintain that equilibrium partly explains why the two men remained in their posts, despite criticism of Johnson's performance as foreign secretary and calls from hard-line exit supporters for Hammond to be fired.
Departures from the Cabinet included Northern Ireland Secretary James Brokenshire, replaced by outgoing Culture Secretary Karen Bradley.
May did get to appoint her own right-hand-man when David Lidington was appointed as Cabinet Office minister to replace Green.
Justine Greening announced she was quitting the Government in protest at being moved from education to work and pensions.
There was little that was dramatic in Theresa May's reshuffle of her Cabinet on Monday, though the British Prime Minister has made the underlying goal obvious enough ~ to add more power to her elbow before the Brexit negotiations are in full flow.
"Social mobility matters to me & our country more than my ministerial career".
The Cabinet changes - at least two secretaries out and seven officials promoted - are May's first major effort to refashion her top team, after months of speculation about her future.
It had been billed as a day in which Theresa May would settle some grievances, and more than a few ministers would end up licking their wounds. Instead, Esther McVey becomes Work and Pensions Secretary.
United Kingdom cabinet reshuffle leaves more questions than answers
Monday's bungled cabinet reshuffle was underlined by the Tory press team wrongly announcing Mr Grayling as the party's new chairman only for Brandon Lewis to be appointed an hour later.
James Brokenshire steps down as Northern Ireland Secretary for health reasons.
Work and pensions minister David Gauke is now justice secretary.
Jeremy Hunt had social care added to his Health Secretary brief, regardless of the crisis gripping the NHS in England under his leadership.
The Tories also received derision after publishing a tweet congratulating Chris Grayling on becoming party chairman - then deleting it seconds later.
Claire Perry will also be able to attend the Cabinet as a new Business Minister.
It's a key role both for Brexit and the stability of May's minority government as it will involve delicate negotiations over the Irish border and devolved government with the Democratic Unionist Party, whose lawmakers prop up the government after May lost her majority in June's vote.
One week later, as worldwide development secretary, after a row over unauthorised meetings with Israeli officials.
David Davis is staying on as Brexit Secretary, while Liam Fox remains as International Trade Secretary.
Terminator provider BPAS said it was "incredibly disappointed" with the decision due to the MP's past opposition to the decriminalisation of abortion.