BBC Radio 5 live's Ian Dennis reported punches were thrown, while the police had to form a "human wall" to protect Liverpool fans in the United section.
Seats were also thrown by supporters.
Greater Manchester Police arrested five men; two on suspicion of affray and one who was cautioned for lighting a flare.
In a statement, superintendent Jim Liggett said GMP was investigating a "handful of concerning incidents".
Following fighting between fans from both sides in a small section of the crowd, the police had to escort a number of supporters out of the ground and on to their coaches.
A number of smoke bombs and flares were let off in the away end shortly before half-time, GMP added.
The game ended 1-1, with Liverpool winning 3-1 on aggregate to progress to the quarter-finals.
Uefa, the governing body of European football, says it is awaiting the referee's report from the game before deciding on what action to take.
Guidelines from Uefa state host clubs are liable for order and security inside and around the stadium.
All clubs may be subject to disciplinary measures for "inappropriate behaviour on the part of their supporters" including the throwing of objects, the lighting of fireworks or any other objects and "any other lack of order or discipline observed inside or around the stadium".
Before the match, police had to remove a banner on the M602 motorway in Salford designed to taunt Liverpool fans heading to Old Trafford.
United were not punished by Uefa after after some of their fans sang offensive Hillsborough disaster chants during the first leg at Anfield.
Manchester United and Liverpool could face Uefa action after fans lit flares and were involved in scuffles during the Europa League last-16 second leg at Old Trafford.