According to the Times, the two leaders will try to restore their “brotherly” relationship.
Mr Johnson and Mr Macron are expected to speak in the coming days after relations hit an all-time low over fishing rights.
A senior Government source told the Times that both the UK and French sides were keen to “dial down the rhetoric” before the G7 summit next month.
The source added: “We’re a bit like a pair of brothers.
“We’re the closest allies and there is no fundamental unhappiness but things are bumpy.”
French fishermen claimed that their rights were unfairly restricted by licenses issued by Jersey’s new system.
About 60 French fishing boats protested at St Helier while two Royal Navy ships and two French vessels were sent to the area.
On Thursday, the protest ended after talks took place to resolve the row.
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