The Red Devils have registered their interest in Bergwijn, 21, after it was reported Bundesliga giants Bayern Munich are weighing up a move for him.
Liverpool and Inter Milan are also thought to have been keeping tabs on the winger in recent weeks.
But United have ramped up their interest in the 21-year-old, sending over a scout to watch the ex-Ajax man in action last Sunday ay FC Emmen.
The Red Devils are thought to have wanted to see their man in action out of his comfort zone of the Philips Stadium.
Bergwijn has impressed once again this season, scoring 10 goals in 17 Eredivisie matches.
But the Holland international has more than one string to his bow, also registering 10 assists.
Bergwijn will come at a price though, currently valued at around £25m, and without any proven success in Europe or the top leagues.
The winger is under contract until 2022, having agreed a new deal last summer.
“Yes it is tough and intense – but in a good way”
Elsewhere at Old Trafford, Romelu Lukaku has admitted that he is fighting for his place amongst the United forwards.
The £75m Belgian has slipped behind Marcus Rashford in the pecking order in recent weeks, with United boss Ole Gunnar Solskjaer admitting Rashford was his No. 1 choice.
Having been on the bench for the last four league games, Lukaku was handed a start in the 3-1 FA Cup win over Arsenal on Friday.
And the 25-year-old admits things have been “tough” recently, but insists he is continuing to fight for the good of the team.
“Yes it is tough and intense – but in a good way. It’s all about the team,” said the Belgian.
“We try to push each other to make sure we win because we know that we are still not in the greatest position in the league but we all want to be in the top positions.
“We want to win a trophy as well. The manager has done a great job so credit to him but to the players as well and the team.
“Everyone wants to play for the team – it’s the team above everything.”