An opinion poll for BFMTV on Thursday suggested 61% of French voters were worried by the political situation.
Macron cut short a trip to Poland on Thursday and had been expected to name his new prime minister last night, but he then postponed his announcement until Friday morning.
He then met Bayrou, 73, at the Elysée Palace for about an hour and three-quarters and a red carpet was seen being prepared at the prime minister’s residence at Hôtel Matignon.
Macron has vowed to remain in office until his second term ends in 2027, despite Barnier’s downfall last week.
If he does pick Bayrou, the challenge will then become forming a government that will not be brought down the way Barnier’s was in the National Assembly.
It is thought he will either seek to bring parties from the centre left into the government, or agree a pact so they do not oust the next prime minister as well.
Macron has already held round-table talks with leaders from all the main political parties, bar the far-left France Unbowed (LFI) of Jean-Luc Mélenchon and far-right National Rally of Marine Le Pen.
Among the other favourites to replace Michel Barnier, who lasted only three months as prime minister, were Defence Minister Sebastien Lecornu and centre-left ex-prime minister Bernard Cazeneuve.
French media quoted another potential candidate, former foreign minister Jean-Yves Le Drian, as saying he had turned down the job because he was in his late 70s.
Former Brexit negotiator Barnier was voted out when Le Pen’s National Rally joined left-wing MPs in rejecting his plans for €60bn (£50bn) in tax rises and spending cuts. He was seeking to cut France’s budget deficit, which is set to hit 6.1% of economic output (GDP) this year.
