The official said they're currently working with a range of companies on the standards, including non-profits and tech companies, adding that they are likely still weeks away from being finalized, the official said.
Multiple government agencies are engaged in conversations and planning, coordinated by the White House, as this kind of system will play a role in multiple aspects of life, including potentially the workforce, another senior administration official told CNN.
The Biden administration previously said the federal government should not be involved in efforts to create a vaccine passport system to verify that people have been vaccinated.
The Post, citing five officials who spoke on the condition of anonymity, reported the effort "has been driven largely by arms of the Department of Health and Human Services, including an office devoted to health information technology."
White House coronavirus coordinator Zients has led the White House's role in coordinating the project, an official told the Post.
"Our role is to help ensure that any solutions in this area should be simple, free, open source, accessible to people both digitally and on paper, and designed from the start to protect people's privacy," Zients said earlier this month, according to the newspaper.
The administration official told CNN Monday the standards are still in the early stages of development but expects them to look similar to what Zients outlined -- that the ultimate product should be free, private and safe from being duplicated or forged.
But part of ensuring wide usage for vaccine passports is accounting for the large subset of the global population that still doesn't use or have access to smartphones. A few companies within the Covid-19 Credentials Initiative are also developing a smart card that strikes a middle ground between the traditional paper vaccine certificates and an online version that's easier to store and reproduce.
This story has been updated with additional information on the government's involvement in coordinating vaccine credentialing standards.
CNN's Rishi Iyengar, Kevin Liptak and Chandelis Duster contributed to this report.