The Brussels bureaucrat is understood to be under consideration by President Emmanuel Macron for the role. The pair became close political allies throughout the negotiations over Britain’s European Union exit. And Mr Barnier last week said he would be on the hunt for a new job at the end of the year after trade talks with the UK are wrapped up.
The Frenchman told a think-tank event that “somebody else” would “continue the negotiations on the other issues” in the UK and EU’s future relationship after a free-trade agreement is concluded.
Mr Macron is currently looking for a replacement for current prime minister Edouard Philippe, who was recently elected as mayor of the northern port city of Le Havre.
The President is said to want to recruit a trusted deputy in order to seize greater control of France’s executive.
Mr Philippe has currently appointed a temporary official to carry out his mayoral duties until a successor is found for his high-profile role in Paris.
Mr Macron’s cabinet is expected to discuss filling the job when his senior ministers meet later this week.
The inclusion of the EU’s chief Brexit negotiator on the list of prime ministerial candidates hands an unlikely boost to Boris Johnson’s bid to deliver an agreement with Brussels by the summer.
Ahead of this week’s Brexit talks, Mr Johnson had already announced that his negotiator, David Frost, had been handed a significant promotion.
Mr Frost will become the Prime Minister’s National Security Adviser in August, replacing outgoing Civil Service chief Sir Mark Sedwill.
European sources were encouraged by the announcement, claiming it had raised hopes of a swift Brexit deal.
Mr Barnier’s spokesman said Mr Frost’s new role would not impact the negotiations.
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During last year’s discussions over the next European Commission president, Mr Macron was ready to support Mr Barnier in becoming the bloc’s top eurocrat.
But EU leaders opposed his nomination as some feared it may have handed the French President too much power in Brussels.
Instead, the bloc appointed Ursula von der Leyen to replace Jean-Claude Juncker and Mr Barnier carried on as the EU’s lead negotiator with the UK.