The Brexit Secretary gave the European Commission’s chief negotiator a first edition of one of the great mountaineering books – a French language version of Regards vers l’Annapurna signed by French author Marcel Ichac.
It is a stunning account of an epic French expedition to the Himalayas in 1950.
Mr Barnier gave Mr Davis a hand-carved wood walking stick from his home town, Savoie in the French Alps.
A UK source said: “Both gifts reflected their shared love of climbing and mountaineering.”
In May, Mr Barnier issued a thinly-veiled warning to Prime Minister Theresa May about the perils of Brexit after discovering their shared love of hill walking.
Mrs May called a snap general election after a walking holiday in Snowdonia in April.
Mr Barnier said: “If you like walking in the mountains, you have to learn a certain number of rules.
“You have to learn to put one foot in front of the other, because sometimes you are on a steep and rocky path.
“You also have to look at what accidents might befall you, falling rocks. And you have to keep looking at the summit, the outcome.”