This is highly significant, because a "full, proper public inquiry" means one led by a judge and with witnesses represented by lawyers.
I am also told - though Downing Street is refusing to comment on this - that the Cabinet will be asked by the prime minister to approve the terms of the inquiry on Wednesday morning, and there could be an announcement shortly afterwards.
Such a public inquiry - like Leveson's into hacking and Chilcott's into the decision to go to war in Iraq - would take many years and might not report until after the next election.
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During the Queen's Speech debate, in answer to a question from the Lib Dem leader Sir Ed Davey on the timing of the start of an inquiry, the PM said: "I can certainly say we will do that within this session... I do believe it's essential that we have a full proper public inquiry into the Covid pandemic".
Although the PM has repeatedly said there would be an inquiry, he has never indicated when it would start and has never suggested whether it would be a "full, proper public inquiry" or a less formal and less time-consuming "independent" one.
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According to a source, the prime minister has decided to confirm the timing of the inquiry so that he is seen to be taking the initiative, rather than reacting to potentially damaging disclosures due to be made by his estranged former chief aide Dominic Cummings, who is giving evidence to MPs on the health and science committees on May 26.
He will tell MP that the prime minister was too slow to lock down in March and rejected his insistence that there should be a second lock down in September.
Cummings believes this slowness to restrict our freedoms led to significant increases in the death toll.
However, the prime minister is understood to be confident that voters understand his reluctance to lock down and that they will give him credit for the success of the vaccine programme.