United were comprehensively beaten by the Catalan giants in the Champions League final for the second time in three years when they went down 3-1 at Wembley in May.
Another level: Lionel Messi surrounded by six United players during Champions League final
Neville was part of the United side that overcame Juventus - who they had lost to three out of four times in the previous two seasons - on the way to claiming the 1999 European Cup, and the 36-year-old believes Barcelona will also be caught.
'We had young players who strived to get to their (Juventus') level and eventually you get there,' the former Red Devils defender and captain told the Manchester Evening News.
'You climb that mountain and it is tough. But I think we will progress and we will get there.
Italian job: Neville (right) has seen United rise to the challenge from the likes of Juventus
'I think the likes of Man United and Real Madrid will get past Barcelona in the next two years.
'Next season will be tougher for them because teams like United and others will learn. Things go in cycles.'
Paul Scholes: Arsenal may play the prettiest football, but they flatter to deceiveSunderland aim to secure United trio as Bruce bids to continue spending spreeAll the latest Manchester United news, features and opinion