The plaintiffs and others Monday asked that the ruling issued in late July not be delayed and that the case be reviewed as quickly as possible.
Virginia officials are supporting a delay to minimize disruptions while the issue is resolved.
Unless the court intervenes, same-sex marriages could begin Thursday. The state also would need to start recognizing out-of-state gay marriages.
The request will either be considered by Chief Justice John Roberts or the full court. It has twice granted delays in related cases.
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