Mr Johnson was accused of lying during the EU referendum campaign by Mr Ayrault, who then said the foreign secretary had behaved with “humility” at his first EU summit in July.
Prime Minister Theresa May will continue Brexit talks with EU leaders.
Mrs May will travel to Slovakia for talks with Prime Minister Robert Fico before travelling on to Poland for a meeting with her counterpart Beata Szydlo.
The meetings are the latest in Mrs May’s diplomatic push ahead of formally triggering the two-year process of withdrawing from the EU – expected to begin early next year.
The prime minister has already met Italian PM Matteo Renzi, Irish PM Enda Kenny, German Chancellor Angela Merkel and French President Francois Hollande.
Mr Johnson’s visit to France on Thursday is his first official European bilateral meeting with Mr Ayrault, who has openly criticised the new foreign secretary.
Mr Ayrault accused Mr Johnson, who led the campaign to get the UK out of the EU, of having “lied a lot” to the British public during the referendum campaign and said he had “his back against the wall to defend his country”.
Reacting to the comments, Mr Johnson said it was “inevitable there will be some plaster coming off the ceilings in the chancelleries of Europe” but that “they are making their views known in a frank and free way”.
He also said he had received “a charming letter” from Mr Ayrault saying “how much he looked forward to working together and deepening Anglo-French co-operation”.
After meeting at an EU Foreign Affairs Council in Brussels earlier in July – Mr Johnson’s first summit since taking up the post – Mr Ayrault said his British counterpart had arrived with “some humility”.
The pair are likely to discuss the recent terror attacks in France, as well as the bilateral relationship and defence and border control issues.
Mr Johnson’s remit does not specifically cover negotiations on Britain’s exit from the EU, as Mrs May appointed a new post of Brexit secretary, taken up by David Davis.
Mr Johnson has insisted that Brexit does not mean Britain will be leaving Europe, just “leaving the EU”.