"We want to assure anyone who left empty-handed that new stocks are coming in the next days," an Aldi spokesman told the Bild newspaper.
The German government is relying heavily on rapid tests to steer country through the next phase of the pandemic as citizens tire of coronavirus shutdowns but the pace of vaccinations remains sluggish.
From Monday, all Germans will be entitled to one free rapid test a week, carried out by a professional at pharmacies or designated testing centres.
Several manufacturers have also received regulatory approval for DIY tests meant for home use.
The goal is to give people more freedom after months of closures and limits on socialising, but experts have stressed that the quick tests are less reliable than standard PCR testing and that the usual hygiene precautions should be maintained even with a negative result.
Aldi is limiting the purchases to one packet per person, containing five tests for 24.99 euros ($30). The test is done with a nasal swab and delivers a result within 15 minutes.