British number five Ward, 29, lost 6-4 6-3 to Italy's Matteo Donati, 21, a player ranked 99 places below him in the world at 270.
Klein, 26, was beaten 6-3 6-3 by Ukraine's Sergiy Stakhovsky, 30, ranked 96th in the world and 115 places above Klein, Britain's sixth-ranked player.
Britain has no-one else in qualifying.
Andy Murray, Aljaz Bedene and Kyle Edmund are in the main draw of the men's event, which starts in Paris on Sunday.
The British quartet of Johanna Konta, Heather Watson, Naomi Broady and Laura Robson are in the women's singles.
Meanwhile, Ward, Edmund and Dan Evans have been awarded wildcards for the Aegon Championships at Queen's Club next month.
All three featured in GB's Davis Cup winning team last year, with Evans and Edmund both now top 100 players.
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The British pair of James Ward and Brydan Klein both failed to reach the French Open after opening-round qualifying defeats at Roland Garros.