The two European giants will face off in the quarterfinals of Europe's major club competition, with the first leg slated to take place in Spain on Tuesday.
Speaking about the Premier League champions' chances, McAteer said:
"I think Liverpool can get past Madrid in the Champions League; they have always got a big performance in them. Liverpool have got a love affair with the competition. I think it's a welcome relief when it comes around because of the negativity that they face in the Premier League."
''Real Madrid are in a good run of form, so you've got to be very careful of them, you've got to respect them. There's unfinished business from 2018 when they beat us in the Champions League Final and I think that will come into play, that will be in the team talk."
The 49-year-old added:
"Madrid aren’t the team that they were that night and I think Liverpool have got a realistic chance of beating them over two legs.
"Klopp’s very good at masterminding two-legged affairs. He's only lost one of his time at Liverpool, that was to Atletico last season."
McAteer also believes the Anfield outfit will secure a Premier League top-four finish this season, having returned to form after a testing start to the year. He said:
"Absolutely Liverpool can make the Top 4. People always say, ‘Chelsea's run-in looks more difficult, United's is difficult, and Liverpool's is easy’.
"I'm not really buying into that. Having played the game, as the season comes into the final 10 games it's where it becomes really mentally and physically tough.
"Of course, there's difficult games for Liverpool coming up. Fulham are still looking at getting out of that bottom three. There are teams that are going to be desperate for points who are fighting for their lives down there who are going to offer Liverpool a challenge.
"I do think the recent international break will have done Liverpool good. Players will get a lot of rest, there's players that have stayed home and not gone on international duty. The rest will do them good."
Liverpool will return to domestic action after their trip to Spain. They will host Aston Villa on Saturday.
Key absences in defense set to impact Liverpool and Real Madrid clash
Both Liverpool and Real Madrid will be without the services of their first-choice central defensive partnership when they take to the field on Tuesday.
The Reds have long since adapted to life without their first-choice defenders, having played in the absence of Virgil van Dijk and Joe Gomez for most of the season.
Real Madrid had seen captain Sergio Ramos return from a one-month injury layoff only for him to suffer a calf injury while on international duty.
Los Blancos were further rocked by the news that star defender Raphael Varane tested positive for COVID-19 on Tuesday morning, which will see him enter self-isolation.