The number of assisted deaths could be more than 4,000 in the 10th year after the law comes into force in England and Wales if MPs vote for it, a review of the policy says.
The estimate forms part of the official impact assessment, carried out by civil servants to inform MPs as they debate whether to allow assisted dying.
It said initially the numbers coming forward would be low, but are expected to grow over time.
The impact assessment says that the total number of assisted deaths is estimated to range from between 164 and 787 in the first (half) year of the service to between 1,042 and 4,559 in year 10, and assuming a start date to the policy of October 2029.
The upper estimate of just over 4,500 assisted deaths would still only represent less than 1% of all deaths, the document said.