Solskjaer was in the starting line-up for United’s semi-final victories over the Gunners in 1999 and 2004.
He went on to lift the trophy with the Old Trafford club on both occasions.
Arsenal will host Manchester United in the FA Cup fourth round and holders Chelsea will face Sheffield Wednesday or Luton for a place in the fifth round.
Manchester City, who beat Rotherham 7-0 in the third round, will need to overcome Premier League rivals to keep alive their FA Cup dream.
Gillingham’s reward for beating Cardiff is a visit to Swansea, while National League Barnet were drawn against Championship Brentford.
The Bees were the only non-league side left in the draw, alongside 14 Premier League teams.
Monday night’s draw confirmed Arsenal as the home team.
It again recreates the classic match-up in the final 40 years ago which Arsenal won 3-2 thanks to Alan Sunderland’s late goal.
Other notable ties include two all-Premier League clashes — Crystal Palace against Tottenham and Manchester City versus Burnley.
Maurizio Sarri’s Chelsea, who won the FA Cup under Antonio Conte last season, host either Championship Sheffield Wednesday or League One Luton Town.
Non-League Barnet, the lowest- ranked club left in the competition, were handed a home tie with Brentford.
Manuel Pellegrini’s West Ham will face AFC Wimbledon, the League One side they beat 3-1 in the Carabao Cup third round in August.
That saw Pellegrini secure his first win as Hammers boss.
The winner of the replay between Newcastle and Blackburn takes on Watford at home.
Everton face a trip to London to play Millwall.
FA CUP FOURTH ROUND MATCHES
Swansea v Gillingham
AFC Wimbledon v West Ham
Shrewsbury or Stoke v Wolves
Millwall v Everton
Brighton v West Brom
Bristol City v Bolton
Accrington v Derby or Southampton
Doncaster v Oldham
Chelsea v Sheffield Wednesday or Luton
Newcastle or Blackburn v Watford
Middlesbrough v Newport
Manchester City v Burnley
Barnet v Brentford
Portsmouth v QPR
Arsenal v Manchester United
Crystal Palace v Tottenham
Ties to be played January 25-28