The Chilean missed out on European glory during his time at Barcelona, who won the title in 2011 — prior to his arrival from Udinese — and again in 2015, just after he left for Premier League side Arsenal.
Sanchez, who is looking to turn the corner under interim manager Ole Gunnar Solskjaer, is hopeful about ending his Champions League drought with his current club.
"The Champions League is a dream for any footballer," Sanchez told the Mirror ahead of Tuesday's first leg against PSG in Manchester, as quoted by Goal.
"I lived with that dream at Barcelona and saw what it was like. I hope I can win it one day and why not here?
"United are a club with so much history and are capable of beating any rival.
"I think we can do that and I see this United team with a lot of chances to score in Paris," the 30-year-old concluded.