The Reds finished eighth in Klopp's first season and fourth in the following the campaign, while the team also reached the League Cup and UEFA Europa League finals in that time.
Barnes comments comes after FA chairman Greg Clarke resigned this week for calling black footballers 'coloured'.
"You can put structures in place, but until you change your perception, nothing will change,' Barnes told PlayOJO.
"I use Klopp as an example,' Barnes continued. 'How successful was he really in the first two years in terms of being 25 points off the top? But we [Liverpool] believed that he was the right man.
"Under different circumstances, he would have lost his job. If he was black, he would have lost his job in those first two years."
Barnes also feels that the top six clubs in the Premier League 'discriminate' against English managers.
"If he [Klopp] was English, he would have lost his job,' said Barnes.
"This is where the idea of discrimination is a little bit strange because there's a group of people who are discriminated against in English football and the very highest level in the Premier League.
"Now, the top six teams will not have a white English manager. Yes, you can say Frank Lampard, as people have pointed out, but that's because he has a special bond with Chelsea and that's the only reason he managed Chelsea - but under different circumstances he wouldn't have got that job."
