McLaren and engine partner Honda have agreed to let Alonso take part in the Indianapolis 500 event on May 28.
Director Zak Brown has insisted that no replacement has yet been found for the Spaniard, but BBC Sport claims that Button - contracted to the Surrey-based team as a reserve driver - will step in barring any unexpected circumstances.
Button, crowned world champion in 2009, retired from F1 at the end of last season and is currently training for Ironman triathlons in California.
The 37-year-old tweeted on Wednesday, "Why do I have so many missed calls?" in an apparent reference to the news that Alonso would be absent from the Monaco Grand Prix.