United sneaked past Galatasaray at Old Trafford in their opening match of the group stage, winning 1-0, but have not looked too convincing in recent European competition and failed to progress to the tournament's knockout stages last season.
Cluj beat Basel, who consigned United to the Europa League in 2011-12, over two legs to qualify for the first round proper and saw off Braga in their opening fixture.
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Brazilian midfielder Rafael Bastos scored twice in the triumph against Braga and Hagi identified the 27-year-old as one to watch at the Stadionul Dr. Constantin Radulescu.
"He's an example of one of their strong foreign players and of their confidence," Hagi claims. "CFR were not a big club in Romania until five years ago when they started winning trophies.
"They've played in the Champions League before and held Chelsea at home so they won't be afraid of Manchester United."
"They're well organised by an experienced coach, Ioan Andone, and while they're not at the top of the league because of injuries and the responsibility of playing in the Champions League, they're a very strong side."
Hagi, who is one of a select group of players to have played for both Real Madrid and Barcelona, had fond memories of facing United in European competition during the 1990s and revealed his admiration for talismanic forward Eric Cantona.
"Cantona played for the crowd," Hagi explained. "That's what I tried to do because the crowd is the most important thing in football.
"It was a shame that he couldn't play for Manchester against Barcelona when I was there in 1994, because Barca won 4-0. [Peter] Schmeichel had to sit in the stands too. He was the best goalkeeper of the 90s, so strong and flexible for a man of his size."
The 124-time Romania international's respect for Sir Alex Ferguson's team has not faded since then.
"Ferguson is an example to every coach in the world," Hagi enthused. "[Ryan] Giggs is a wonderful player who was about skill and imagination, not power. He isn't like a typical British player.
"United are very respected in Romania, as a team who usually reach the Champions League semi-finals or finals should be."