It means the Prime Minister is having to perform a shake-up of his top team during the second national lockdown in England and with post-Brexit trade negotiations growing ever closer to the deadline.
Here, we take a look at some of those who could be in line for the job.
Some senior Tories have been urging Mr Johnson to appoint David Canzini. He is a close ally of Tory elections guru Lynton Crosby and helped run Mr Johnson’s successful Tory leadership campaign.
He was head of campaigns for the Conservatives before joining Sir Lynton’s firm CTF Partners.
Mr Canzini advised Eurosceptic Tory MPs on the “Chuck Chequers” campaign, which lobbied against former PM Theresa May’s Brexit deal blueprint.
He also previously worked for the Scottish Conservatives and David Davis’s 2005 leadership campaign and started his political career as Edwina Currie’s electoral agent in South Derbyshire.
Lord Feldman of Elstree is a former Conservative party chairman who was responsible for fundraising David Cameron’s successful Tory leadership campaign in 2005.
Mr Johnson has called already him to ask him to take on the role, according to reports.
However, it is understood Lord Feldman of Elstree turned the offer down because he is happy with life outside politics.
Australian political strategist Isaac Levido ran the Tory general election campaign in 2019 and is the protégé of Lynton Crosby.
He was widely credited as being the force behind the Tory landslide victory in December.
Inside Conservative HQ, officials reported “Oh Isaac Levido!” to the tune of Seven Nation Army when the scale of the election victory was revealed.
He previously worked on the 2015 and 2017 general election campaigns before returning to Australia to serve as Liberal party deputy director from January 2018 to mid-2019.
Others suggested include Mr Johnson’s foreign policy adviser John Bew, Vote Leave veteran Paul Stephenson and former special adviser Gabriel Milland.
Tory Big Beasts
Some of the senior Tories being touted include:
Sajid Javid was Boris Johnson’s Chancellor until February when he resigned after refusing a demand to fire his team of aides.
Mr Javid also served as Home Secretary under former Prime Minister Theresa May.
He helped to draft Conservative Party manifesto and is currently the MP for Bromsgrove. It is thought he could win support for tax rises from sceptical Tory backbenchers.
Greg Clark has served as a Government minister for nine years continuously – including five in Cabinet.
Most recently he served as Business Secretary under Theresa May.
The MP for Tunbridge Wells is reportedly seen as exceptionally experienced in Government by Boris Johnson allies in the Commons.
Sir Michael served as Defence Secretary under both Cameron and May from 2014 to 2017.
However, he stood down during the Westminster sexual misconduct scandal after admitting that his behaviour towards women in the past had “fallen short”.
He served as the MP for Sevenoaks in 2019.
Sir Michael also served as deputy chairman of the Conservative Party and held ministerial roles in the Treasury, Education and Energy departments spanning four prime ministers.