McLaren have not qualified higher than seventh on the 20 car grid in five races this season.
Button, the 2009 world champion and former Monaco winner, is standing in for Fernando Alonso in the Mediterranean principality while the Spaniard competes in the Indianapolis 500 on the same day.
The penalty will only confirm in Alonso's mind that the double world champion, who has been the centre of attention at Indianapolis, was right to miss Monaco for a shot at glory across the Atlantic.
McLaren are the only team yet to score a point this season, the former champions beset by a lack of reliability and performance.
Monaco had represented a real chance for the team to change that, however, with the tight and twisty street circuit rewarding driver skills and making engine power less of a factor.
Button, who had not driven a Formula One car for more than six months until he stepped into the cockpit for Thursday practise, was 12th in Saturday's final session before qualifying.
Belgian team mate Stoffel Vandoorne, who already had a three place penalty carried over from the previous race in Spain, was 10th.
Button, 37, had said on Thursday that he was enjoying being back behind the wheel.
"I definitely stepped away at the right time and I haven’t missed driving a Formula One car. But today I really enjoyed it, more than I have for a couple of years to be fair," he said at the time.
"I'm enjoying it very much, probably more than I thought I would actually."
(Editing by Amlan Chakraborty)
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