Murray fell behind 5-0 in the opening set, and although rallying to 5-4, Frenchman Barrere held his serve to win the opener.
The 77th-ranked Briton was broken to love in the third game of the second set and fell behind 3-1 after failing to convert two break points in the following game.
Barrere, rated 115th in the world, led 5-2 before serving out to win the match in one hour and 27 minutes.
Murray took three of six break points, while his opponent converted six of seven.
Gregoire Barrere defeated Andy Murray 6-4 6-2 to knock him out of a clay-court ATP Challenger event in Bordeaux.
Murray fell behind 5-0 in the opening set, and although rallying to 5-4, Frenchman Barrere held his serve to win the opener.
