Turkey international Calhanoglu, who has garnered a reputation as one of Europe's deadliest free-kick specialists, was watched by scouts from Chelsea prior to the Bundesliga's winter break, according to the Daily Mail.
Representatives from Arsenal, Tottenham Hotspur, United and Everton were also reportedly in attendance. Eyes may also have been on Germany prodigy Julian Brandt, who has also been linked with a move to the Premier League.
The 22-year-old Calhanoglu has also been tentatively linked with a move to Barcelona in recent seasons, and he has revealed there are two clubs he would find it very hard to say no to.
"If there was offers like everyone I would be happy. It would be very hard to decline an offer. My job is open to anything, he told LigTV. "Barcelona and Manchester United are both big clubs."
United boss Jose Mourinho has played down talk of a busy January window, but admitted he could possibly make one addition to his squad.
"I keep saying the same, I like my squad. I believe more in the summer market than in the winter window. More time, more choice, I think it's better. But let's see what happens," Mourinho told Sky Sports in December.
"If we buy a player we buy a player, we're not buying two, three or four. If we open the door for someone to leave it's not because we push him, it's because he really wants to leave."
That player was thought to be Benfica centre-half Victor Lindelof, but United's interest in the Sweden international cooled over the festive period.
United could be busier on the outgoing front, although Mourinho has insisted only one member of his squad will be leaving on loan in the shape of Sam Johnstone, with his move to Aston Villa confirmed on Thursday.
Memphis Depay and Morgan Schneiderlin have both been told they can leave Old Trafford this month, but only if United receive satisfactory transfer offers for the pair.