It is understood that Morgan is considering scrapping the building and staying at the assembly’s temporary home, converted office block Crickhowell House, or moving to different premises in Cardiff.
A spokesperson for the assembly said: “The project has been put on hold while the first secretary reviews the current situation. It has not been shelved; it is just on hold.”
The announcement comes as the assembly is due to vote on committing an additional £1m to take the design forward.
The assembly has already spent £1m on the building’s design. The overall cost of the building has almost doubled since it was first conceived.
Work was scheduled to start on site in November. However, no contractors have been appointed to the scheme. Richard Rogers Partnership declined to comment.