Swansea have survived relegation from the league on the final day of the season in years gone by and also played in Europe.
But Monk said: "United have fantastic players and a great history, so it's the biggest game we've had in the club's history.
"United have been champions for a record number of times now and are one of the biggest clubs in Europe and around the world. They are recognised on a par with Barcelona and Real Madrid, but they set the standard for British football.
"It's one everyone connected to the club - the players, the fans and the city - are looking forward to. It will be a great occasion for everyone.
"It's something we can all be proud of and if we can get a result, it'll be the icing on the cake. But we're at home, doing well and at our best we'll give anyone a game."