The youngsters are claimed to be targets for Ole Gunnar Solskjaer at Old Trafford as the Norwegian plots to bring in young talents ahead of his first transfer window as permanent manager.
United will likely look to recruit midfielders, with speculation continuing to surround the future of Paul Pogba after it emerged the France international is keen on a move to Real Madrid.
His team-mate Ander Herrera also looks likely to move on when his contract expires at the end of the season.
Those reportedly iminent departures have left the Red Devils eyeing Turkish winger Omur, who has scored four goals and notched seven assists in Super Lig this season and is seen as one of the most promising players in the division at 19 years of age.
And the Premier League side are also considering 22-year-old midfielder Yazici, who has managed three goals and two assists, having made a breakthrough season for Trabzonspor last year.
Lille are also expected to dip into the market this summer after the Ligue 1 club's chairman admitted he expects the highly-rated Nicolas Pepe to move to a big European club.
When asked by reporters if scouts were present for Trabzonspor's 4-2 win over Kayserispor on Monday night, Agaoglu said: "Our players have been watched closely for a long time.
"Today, Manchester United and Lille scouts were at the game.
"It makes us very proud that Trabzonspor players are on the radars of the biggest clubs in Europe."
Both the players who were being watched got on the scoresheet as Trabzonspor cruised to victory over their 13th-placed opponents.
The Super Lig side still have an outside chance of making the Champions League qualifiers for next season.
They currently sit in fourth, but must overhaul a seven-point gap to second-placed Istanbul Basaksehir in the final three games of the season.