The men, trapped underground for two months, were freed in October.
Sir Bobby, whose father was a miner in Northumberland, told the Chileans: "You took it in your stride and the whole world was very proud of you."
Omar Reygadas said they were delighted to be at Old Trafford and jokingly challenged United to a game.
The 56-year-old bulldozer operator said: "It's an honour to be at Old Trafford. It's the theatre of dreams and we have only seen it on television.
"We are very glad and excited to watch the match against Arsenal tomorrow," he added.
Then Mr Reygadas joked: "We have got enough footballers to make a team and maybe play Manchester United."
The 26 miners, who include former professional footballer Franklin Lobos, were invited to the game by Sir Bobby, a director and ambassador for United.
He said: "In 1962 I played in the World Cup in Chile and we enjoyed it and I said that if any of them wanted to come back to Manchester they'd be welcome. I had heard some of them were United fans."
The men will be guests of honour at Old Trafford for the game against Arsenal on Monday night.
They will be given a tour of the ground and have a chance to meet the players before the crucial Premiership game.
The 33 miners were trapped by a rock fall on 5 August at the San Jose gold and copper mine.
One by one, they were brought back to the surface in a capsule after spending two months underground.
Since then they have been invited to a number of events abroad and four of them visited Spain and watched a Real Madrid match.
bbc.co.uk/news/
Chilean miners welcomed to Manchester by Bobby Charlton
12.06
Madong Arizona