Scottish Natural Heritage has submitted a report to ministers outlining the options on the future of the animals, which were discovered living on Tayside.
But the cost of continuing to find and trap the animals could be too high and the beavers, which have been living in the wild for up to a decade, may be left alone.
Environment minister Stewart Stevenson is expected to decided imminently whether that course of action can be taken or, as they are an illegal species under the Wildlife and Countryside Act, they should be captured.
The beavers have proved elusive. A previous trial attempt by SNH to trap the animals resulted in just one animal being caught, despite months of trying. The beaver was taken in by Edinburgh Zoo but later died.
A third option being considered would be to leave the beavers alone while the high-profile trial reintroduction of another group of imported beavers in Knapdale Forest, Argyll, takes place.
Only once the trial is complete and a decision taken on whether beavers are to be permanently reintroduced to Scotland would the minister pass judgement on the wild Tayside beavers.
It is not known how the Tayside beavers made it into the wild, but it is assumed they either escaped from private collections or were released.
There have been wildly varying estimates as to the numbers of the beavers, which are in a stretch of about 30 square miles between Aberfeldy, Forfar, Dundee and Perth, but latest estimates suggest that there are about 100.