United saw off AC Milan 2-1 on aggregate, with Arsenal beating Olympiacos 3-2.
It was less good news for Arsenal’s north London rivals Tottenham Hotspur who dramatically crashed out to Dinamo Zagreb, losing 3-0 on Thursday night to lose 3-2 on aggregate.
Scottish Premiership champions Rangers also saw their run brought to a halt by Slavia Prague.
There is no country protection or seeding for Friday’s draw, which takes place at 12pm GMT, meaning Arsenal and United could face off in the final eight.
They could also face any of the other remaining six teams, including Spurs’s conquerors Zagreb.
Who is in Europa League quarter-finals draw?
Manchester United, Arsenal, Dinamo Zagreb, Granada, Roma, Slavia Prague, Villarreal, Ajax
Granada, Roma, Slavia Prague, Villarreal and Ajax are the other sides in the draw.
Each side will also learn their potential semi-finals opponents on Friday, with the draw being made for the next two stages of the competition.
Manchester City, Liverpool and Chelsea could face any of Real Madrid, Bayern Munich, Porto, Paris Saint-Germain or Borussia Dortmund as well as each other.