The Denmark international was at fault for Lionel Messi's equaliser at Stamford Bridge on Tuesday as his loose pass was picked off by Andres Iniesta.
It was a rare error in an otherwise impressive season from the 21-year-old, who has started 25 matches in all competitions.
Conte claimed he had already forgotten the incident and tipped Christensen to eventually inherit the Blues' armband.
"During your career you can make a lot of mistakes. If I had to speak about my mistakes from my career we would be here all day," the Italian told reporters on Friday.
"Christensen is having a fantastic season and showing great maturity.
"For me, apart from his mistake, his performance [against Barcelona] was wonderful. I trust Christensen.
Conte says Andreas Christensen has been excellent this season and adds he can be a defender for Chelsea for many years, and a future captain, too. #MUNCHE— Chelsea FC (@ChelseaFC) February 23, 2018
"I think this player is a present player for Chelsea and can be here for 10, 12 years. I think he can be captain."
Christensen signed for Chelsea in 2012 and has won 14 caps for Denmark.