The 18-year-old has a €6 million (£5.2 million) release clause in his contract, and Spanish media reported he told the club this week he plans to depart in January rather than renewing his contract.
Flick said he did not want to speak about the issue but indicated he thought people advising the player were blame for his decision, with Paris Saint-Germain and Borussia Dortmund among teams linked to Dro.
"I don't want to answer... as a coach you give a lot of energy to the players, you give them the confidence, you believe in them, you help them to grow and improve of course," said Flick.
"But then there are people around this player. I want to wait until it's done. If they say 'now I go to another club', then ask me again. But not now."
Barcelona's La Masia academy is considered the best in the world and a string of stars have emerged from it to form part of the current squad, including Spain internationals Lamine Yamal and Pau Cubarsi.
Flick issued a thinly-veiled message to Dro, who has played eight times for the club, and other youngsters who may have their heads turned in the future.
"What I want to say to the young players from La Masia, is we are Barça and one of the best teams in the world," continued the coach.
"We give them the opportunity to train with us, to grow, every day, with the best players in the world. We give them the opportunity, we give them the support, believe in them, the confidence.
"If you want to play for Barça, then (do it) 100 per cent with your whole heart. This is what I want to say to everyone who is now with us, or in the future with us: this must be 100 per cent.
"These colours, you have to live for them, this is what I want to see. All the others I don't want, this is what I can say."
The German coach suggested Dro, whom he handed his debut in September, would be making a mistake by leaving Barça.
"Sometimes you do not agree with what they are doing, but they are 18, they are old enough to make the decisions, they have people around them... now I don't want to talk more, it's enough," added Flick.
Waiting on Rashford
Flick said no decision had been taken over whether the club would make Marcus Rashford's loan from Manchester United permanent in the summer.
The England international has impressed this season on Barça's left flank after being frozen out at Old Trafford.
"(Sporting director) Deco and I always speak about our team, what we can do... Marcus's performances have been really good until now, but we have to manage it, it's Deco's job and the club's job to do things for next season, but we have to wait," said Flick.
"We have more time, many months, many days, many weeks, it's like that."
Another player linked with an exit from Barça is goalkeeper Marc-Andre ter Stegen, who has fallen behind Joan Garcia and Wojciech Szczesny in the pecking order.
The veteran German is poised to go on loan to neighbours Girona until the end of the season.
"It's Marc's decision... he's our captain, he's an idol of this club over the last 10, 12 years, it's his decision, we'll accept what he wants," said Flick.
La Liga leaders Barcelona visit Real Sociedad on Sunday, and Flick said winger Raphinha was a doubt for the game with a knock, after the Brazilian did not train with his teammates on Saturday.