The two showpiece races take place on May 28 and McLaren - whose fortunes have plummeted in F1 - "will enter a single auto in the 101st Indianapolis 500", said a team statement.
McLaren's reserve, Button has been drafted in from his sabbatical for the sixth round of the 2017 F1 season, which Alonso will miss to contest the Indianapolis 500.
Renault's Nico Hulkenberg, a Le Mans victor when that sportscar race did not clash with anything, said his situation had been completely different to Alonso's at McLaren.
The former Formula One champion, 37, retired at the end of last season but apparently it took just a two minute phone call to persuade him to race in place of the 35-year-old Spaniard - who is unavailable as he's travelling to the United States to take part in the Indianapolis 500 this weekend. "His auto is going to run with some guy everybody will be pretty familiar with".
"It also reflects the technical versatility of McLaren and Honda and it underlines the fact that we're racers, above everything else". "We've agreed to talk about the future in the summer and that's not changed", Brown added.
"I rang him, we just discussed it and his first reaction was 'great, I'm so excited!' It was an easy discussion to be honest, very straightforward. It will be hard if we can't".
"I'm looking forward to pitting myself against the unique racing challenge that is the Monaco Grand Prix", he said.
'It's Not Too Late' To Boot FBI Director Comey
President Trump says he will not request FBI Director James Comey's resignation, even though the FBI is investigating Trump's potential ties to Russian Federation . "I don't want to talk about that".
The two-time Formula One champion has made no secret of his disappointment with McLaren's ongoing struggles, the Spaniard having failed to finish each of the opening two races of the season. Despite his advancing age and the ongoing frustrations, Alonso still has a burning ambition to distinguish himself as one of the best drivers of all time. "On the planes, I will try to look at some videos and some races from the past years and try to learn as quick as I can".
"Perhaps if the races didn't clash - or do it at the end of his F1 career, but, obviously, McLaren have got this approach, which is perhaps different to ours. It's not an oval, but I will go to there to see the race behind the scene to have the first touch with the team to have a great atmosphere when Indy comes".
A press conference is planned for Friday in Bahrain, though a McLaren spokesman said he was not in the Kingdom. After all, he's missed only a handful of races since Abu Dhabi a year ago, and he's as fit as a fiddle. I think it is good for Formula One.
"After Russia, I will try to be in Indianapolis for a couple of simulator days and hopefully test the auto - but it's not 100 per cent sure".
Andretti chief Michael Andretti, a former IndyCar champion who raced in F1 for McLaren, said Alonso was capable of repeating Rossi's achievement.
The double world champion will race for the Honda-powered Andretti Autosport Indy Car team, which will be branded as a Mclaren for the iconic American race.