The panel has said it will permit no further delays in reducing prison overcrowding, the main way to improve medical treatment for inmates.
If it rejects the request the Brown administration filed Friday, the Democratic governor is expected to seek a reprieve from U.S. Supreme Court Justice Anthony Kennedy, who oversees appeals from California and other Western states.
A stay could delay prisoner releases by a year while the justices consider California's appeal.
The lower court last week ordered Brown to immediately begin taking certain steps, including expanding good-time credits that lead to early release
