Over the course of the film, Alexei mentions on a couple of occasions that he's fought Captain America in the past. However, it's not clear whether he's lying about it in an attempt to make his past escapades much more impressive than they actually were. Later on, Alexei does ask Natasha if Captain America ever mentioned him, and it doesn't seem like a dishonest question.
During a discussion on Reddit, fans have been debating whether or not the Red Guardian ever actually crossed shields with the Star-Spangled Man, with u/Hickspy noting, "That was brought up twice. I expected it to follow the rule of 3 and the third time he would admit he made it up, but that never came."
Meanwhile, Reddit user u/GeBolt pointed out that the Red Guardian could've fought Captain America, but it might not have been Steve Rogers (Chris Evans). "He could've fought Isaiah Bradley as Cap during the Cold War," they wrote. "The timeline fits with what was mentioned in the movie and the serum would've slowed Alexei's aging." It would be an interesting connection, since "The Falcon and The Winter Soldier" only recently introduced Isaiah Bradley (Carl Lumbly) into the MCU. Plus, Steve Rogers spent the best part of the century in ice when he was frozen in 1945 at the end of "Captain America: The First Avenger" before SHIELD found him in 2011.
That said, not everyone was convinced by the idea, with u/Objective-Menu3158 suggesting, "I'm pretty sure he was just lying. When he was fighting Taskmaster who was using Cap's moves, he was getting beaten."
All this tells us is that Marvel needs to explore Red Guardian's story a bit further to give fans a definitive answer about whether he really went toe-to-toe with Captain America.
