Steve Hansen's trophy-laden reign came to an end after the All Blacks failed to win an unprecedented third successive Rugby World Cup.
Assistant coach Foster is reportedly among the leading contenders to step up and replace Hansen.
Two-time World Rugby Player of the Year Barrett believes Foster fits the bill for the top job.
"He's a very intelligent coach. A great team man and hopefully we can have some continuity going forward," he told the New Zealand Herald.
"Steve's legacy is a very strong one. His win percentage speaks for itself. I think the biggest thing with Steve is the way he makes his players feel. Trust and confidence.
"We know he 100 per cent has our back. As a player that's all you can really ask for. He's taught me a lot along the way and I'm very lucky for that."
Barrett says New Zealand were happy to see South Africa lift the Webb Ellis Cup on Saturday with a defeat of an England side that dethroned them at the semi-final stage.
"To be fair we were probably supporting them. It's nice to see a Southern Hemisphere team and players that we are so familiar with and have friendships with lift the trophy," he added.
"If it couldn't be us, we were encouraging them to win."