The free shares would be given out by early next summer if policyholders agree to the insurer's plan to shed its mutual status and join the stock market, where its total value might be close to pounds 4bn.
The NU customers most likely to qualify for the bonanza are holders of "with profits"-style endowment, life insurance and pension policies. Customers with "unit linked" pension and life insurance savings plans and even basic life insurance policies may also be eligible because they are considered "members" of the mutual insurer.
But the 2 million holders of other Norwich Union insurance policies - including motor, house, contents and private medical - or Norwich Union tax-free PEPs or unit trusts are expected to be excluded.
People who take out policies between now and the expected announcement may qualify for the free shares. Equally it may even be possible after the announcement to take out a policy and qualify.
But the Norwich, which refused to confirm that an announcement would come this week, warned against "carpetbagging" - taking out a policy simply for a windfall.
An NU spokeswoman said: "This is not like putting a few hundred pounds in a building society. These are long-term policies; we are talking about paying in pounds 50 a month for 25 years."
The insurer recommends people take financial advice before taking out a policy. To get a windfall from a pension or life insurance savings plan requires a longer and more serious financial commitment than with a building society savings account. In addition, windfall hunters should bear in mind that policies from different insurers can produce markedly different performances and that many life policies are poor choices as investments. Most people are better off with a PEP than an endowment, for example.
The expected announcement should make clear whether entitlements to free shares will be related to how much money someone has with the NU, as well as how long they have been a member, and whether policies that have recently matured or that will mature over coming months will qualify.
In addition, while NU may leave open the door to carpetbaggers, it will have the right to close the door in future if it thinks too many people are taking out policies for the "wrong" reasons.
Norwich Union had already announced that it was considering demutualisation, and a formal decision has been expected for some time. Other insurers are likely to follow.
One other route that remains open for people wanting free NU shares - or to speculate on other potential windfalls - is to buy a "second hand" policy in the traded endowment market. Here you buy a policy that the original holder no longer wants, take over paying the premiums, and then collect the payout when the policy matures or the original holder dies, if first. The trouble for buyers is that prices of NU policies have in many cases risen to reflect the expectation of a windfall announcement by the insurer. In addition such policies are likely to mean an upfront investment of thousands of pounds.