The tabloid cited unnamed government sources as saying that two ministers were pushing for the change, possibly from the end of September.
At present, Germans are required to present a negative coronavirus test if they want to enter offices, some shops and restaurants and can get tested for free.
Testing sites have been expanded in recent months to include pharmacies.
Scholz and Söder back the plan
Finance minister and the center-left Social Democrat (SDP) chancellor candidate Olaf Scholz and Bavaria's premier Markus Söder have strongly endorsed the measure, according to Bild.
Scholz reportedly supported the measure because progress in getting people vaccinated meant that public funds shouldn't be used to test those who are unvaccinated for free.
Free tests are costing the government €3.4 billion ($4 billion dollars), according to Bild.
Separately, Scholz has reminded the public to get vaccinated.
"We have to convince our friends to get vaccinated. This is a matter that touches every one of us," Scholz told newspapers of the Funke media group.
Politicians call for faster vaccination
German politicians are also actively encouraging people to get their shots as the school year, which typically begins in August, is drawing closer.
SPD leader Saskia Esken and the German Association of Cities have called for vaccination campaigns at schools. Esken said she supported mobile vaccination teams at education establishments.
Thorsten Frei, the deputy leader of the CDU/CSU parliamentary group for law and consumer protection, told Bild that free tests ought to continue for children since they're not eligible for vaccination yet.
Meanwhile, Germany is preparing for federal elections in September and the officer responsible for overseeing them has maintained that all necessary precautions for poll workers have been taken into account.
So far, 51% percent of Germans have been fully vaccinated, but experts say it isn't enough to prevent the fourth wave of the pandemic.
rm/mm (DPA, AFP)