The developers of the vaccine said this month they were encouraged by the immune response they had seen in trials so far and were expecting to publish Phase 1 data by the end of July.
The data are expected to be published by The Lancet medical journal.
More than 100 vaccines are being developed and tested around the world to try to stop the Covid-19 pandemic, which has killed hundreds of thousands and ravaged the global economy.
AstraZeneca's experimental vaccine is probably the world's leading candidate and most advanced in terms of development, the World Health Organization's chief scientist said in June.
The company has signed agreements with governments around the globe to supply the vaccine should it be cleared for use.
Shares in AstraZeneca traded 5% higher by 1415 GMT. There was no immediate comment from the company on the report.
A spokeswoman for Oxford University told Reuters the team was awaiting confirmation from a scientific journal of a publication date and time for the data, but gave no further details. "(We) are not able to confirm when it will be released," she said.
Peston said in a blog post: "I am hearing there will be positive news soon (perhaps tomorrow) on initial trials of the Oxford Covid-19 vaccine that is backed by AstraZeneca."
Researchers in the United States reported on Tuesday that Moderna Inc's experimental vaccine showed it was safe and provoked immune responses in all 45 healthy volunteers in an ongoing early-stage study.
Moderna started its Phase II trial in May and expects to start a Phase III trial on July 27.
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