Rose suffered a shin wound at the end of Sunday's 3-1 victory over Leicester, being withdrawn in the 88th minute, and the club are assessing whether he will even make the squad.
"Tomorrow it's impossible for him to start from the beginning," Pochettino told a news conference. "Tomorrow we're going to assess whether he can be on the bench."
Spurs' other left-back, Ben Davies, is out after missing Tuesday's training session. The Welshman, who trained by himself on Tuesday, has missed the last four matches with a groin problem and is not yet ready to return.
As a result, Vertonghen is likely to step in, as he did when Spurs beat Newcastle 1-0 earlier this month. The Belgium international has regularly played in that role for his country.
Vertonghen, 31, will come up against 18-year-old winger Jadon Sancho, who swapped Manchester City's academy for Germany in the summer of 2017 and has contributed eight goals and 11 assists for Dortmund in all competitions this season.
"I was speaking with some of the guys with the national team and they're impressed," Vertonghen said. "He went there as a young player with not too many expectations for this season but he's done very well, scored goals and lots of assists. I'm impressed by his development and it's always exciting to see a young player do so well."
Pochettino also said he has been unsurprised by Sancho's success at Dortmund and insisted Tottenham were aware of his talent before he moved abroad.
"I'm not too much impressed about him," he said. "We knew very well that he was capable of doing what he's doing now, with game time. We've seen the quality that he was showing in Manchester City or in his national team.
"We were talking in the last few years that he was a massive prospect to be a very good player. Of course he's showing now in Germany that he can cope with the pressure to play in the Bundesliga or in the Champions League, and it doesn't surprise me. His talent is there and he's a very talented player."
Spurs are aiming to reach the quarterfinals of the Champions League for the first time since 2011, having been eliminated in the round of 16 by Juventus last season.
Harry Kane and Dele Alli are still out of action, while Vincent Janssen has not been included in Tottenham's squad for the Champions League knockout rounds. But Eric Dier is available again after fighting off a virus, and Erik Lamela is also ready to face to Bundesliga leaders after missing the weekend's match against Leicester with a back problem.
Pochettino has previously voiced his displeasure at the fact that Spurs have had a day less than Dortmund to prepare for the match, having faced Leicester on Sunday, but he expects his team to be fired up and full of running.
"I think the key now is you need to find your energy in your own motivation," said the manager. "Of course that extra charge is there, and I think we're going to arrive in a very good condition. I think you can feel the energy in the eyes of the players. We were talking in the meeting before and it's completely different.
"If you ask me why, I don't know, because I'm a person whose own motivation is always high, when we play against Dortmund or Barcelona or we play against different games in different competitions. But of course you can feel that it's the Champions League, that it's going to be a massive game with, I hope, a full stadium at Wembley. Of course the energy and motivation is going to be there."
Vertonghen added: "We can beat anyone. We've shown that last year. Over two games we were the better team against Real Madrid, who were the eventual winners of the trophy. That's why it was a shame we couldn't do better last year, because we felt we should have done better. That's why everyone has the feeling we can do it this year."