The Times reported on Saturday that workers at the factory were paid as little as 3 pounds ($3.80) an hour to make clothes for Quiz.
Britain's minimum wage is 8.72 pounds for people over 25 years old and 8.20 pounds for people aged 21 to 24.
Quiz said it was investigating the allegations and that its initial review had found the supplier had used a sub-contractor in direct contravention of a previous instruction. It did not identify the supplier.
Quiz's shares dropped around 20% in early dealings, but by 1335 GMT had recovered to trade up 3.4% at 6.98 pence.
The Times report followed allegations in a July 5 Sunday Times report of poor working conditions at a Leicester-based supplier to online fashion retailer Boohoo, which sent its shares 28% lower last week.
Boohoo said on July 8 it was launching an independent investigation into the allegations and that it was committed to driving up standards where required.
It also noted that Britain's The Gangmasters and Labour Abuse Authority had not taken any enforcement action during its investigation into working conditions at clothing manufacturers' factories in Leicester.
(Reporting by James Davey; Editing by Mark Potter and Jon Boyle)