Dortmund hasn't won in the league since September and has only one win across 11 competitive games. It has fallen 10 points behind Bayern Munich, having previously led the defending champions by five.
Bosz was instructed to "turn every stone" to find a solution, but sporting director Michael Zorc indicated the club was sticking by him.
"No," Zorc said when asked if he'd spoken with potential replacements. "We want to achieve the turnaround with Peter Bosz."
Only goalkeeper Roman Buerki prevented a worse ending for Dortmund, denying Leverkusen's Leon Bailey a late winning goal.
Despite Wendell's first-half sending off for a brutal challenge on Gonzalo Castro, it was host Leverkusen that came closest to scoring at the end.
Kevin Volland scored on a counterattack on the half-hour mark, rounding Buerki for a simple finish in an empty net.
It was a deserved lead for the home side, which had 10 efforts on goal compared to Dortmund's one until Wendell's sending off minutes before the break.
Castro had to go off with a heavily bandaged ankle and was replaced by Shinji Kagawa. It was the second injury-induced substitution for Dortmund, which lost Maximilian Philipp to a right knee injury early on.
"The guys will be out for a long time," Zorc said of their injuries.
Leverkusen stayed compact and defended strongly in the second half, when Andre Schuerrle struck the post.
Volland had another chance in similar circumstances to his goal, but Buerki blocked this time, right before Andriy Yarmolenko equalized for Dortmund.
Leverkusen held on to stay unbeaten across 10 competitive games, while Dortmund's drop in the table continued. The side next faces a trip to Real Madrid in the Champions League.
BAYERN BOUNCES BACK
Bayern, which lost for the first time since Jupp Heynckes' return as coach last weekend, recovered by fighting past promoted Hannover 3-1 to increase its lead to six points.
Thomas Mueller returned after nearly six weeks out injured, and the Germany forward crossed for Arturo Vidal to break the deadlock after a quarter-hour.
Vidal, who already struck the post, scored with a header inside the far post. It was his third consecutive game scoring, confirming his improvement under Heynckes, who asked for more from the Chile midfielder.
Niclas Fuellkrug had the chance to equalize from the penalty spot. His first attempt - which was successful - was ruled out for an infringement by a teammate. Sven Ulreich, who conceded the spot kick, saved the second attempt.
But Charlison Benschop equalized with a header off a corner in the 35th.
Bayern's Kingsley Coman made the breakthrough midway through the half, taking Mueller's cross with his right boot and firing the ball in at the near post with his left.
Robert Lewandowski wrapped it up late with a penalty, ensuring he became the first Bayern player to score in the side's first seven home league games of a season.
Serge Gnabry scored twice - the second a brilliant lob from near the halfway line - in a 4-0 win for Hoffenheim over second-place Leipzig.
Schalke failed to take advantage of Leipzig's slip-up as winless Cologne twice came from behind to earn draw at Gelsenkirchen 2-2.
Werder Bremen beat Stuttgart 1-0, and Augsburg won at Mainz 3-1.
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