Barrett and loose forward Ardie Savea are considered the strongest candidates to succeed former skipper Sam Cane and lead the team to the 2027 World Cup in Australia.
"I guess that's something I'd be open to," Barrett told New Zealand media, adding: "It'd be a huge honour to do that ... but my focus right now is to get back on the park".
The lock's comments followed New Zealand Rugby's announcement that he and veteran hooker Codie Taylor had re-committed to the All Blacks through to the 2027 World Cup.
Barrett said being a part of near-misses at the last two tournaments - the final last year in France and the semi-finals in Japan in 2019 - were motivation for another tilt.
"There's certainly a lot of pain for the last couple of World Cups, personally," he said.
"Scars, I guess, that probably won't go away. That's a big driver to give myself another opportunity to go to a World Cup - if I'm playing and performing well."
Barrett is vital for New Zealand's second row stocks following the departures of Sam Whitelock and Brodie Retallick but has battled a back injury which saw him miss most of the Super Rugby Pacific season for the Canterbury Crusaders.
The Crusaders, who failed to qualify for the playoffs, struggled without Barrett and also missed 85-Test hooker Taylor while he was on a non-playing sabbatical.
Barrett is still not fully recovered from a bulged disc in his back but said he was "tracking alright" ahead of the All Blacks' season-opening Tests at home against England next month.
"I'm still getting a plan together for the next month of how I can get back to rugby," he added.
21°C