New Zealand skipper Vettori missed the last two group games aftering injuring his knee in the 110-run win over Pakistan, while pace bowler Mills sat out the loss to Sri Lanka with a thigh strain.
A New Zealand team spokesman says both players are continuing to recover well and should line up on Friday against South Africa.
New Zealand hasn't played since Friday's loss to Sri Lanka in Mumbai and didn't find out until Sunday where its quarterfinal would be played.
The squad arrived in Dhaka at midday local time on Tuesday and decided not to hold a daily practice session.
As well as being a key bowler for his side, Vettori adds experience to the Black Caps' batting line-up, which was missed as they slumped to 153 all out against Sri Lanka.
"We need to show more application to stay at the wicket," coach John Wright was quoted as saying. "We've got the shot-making but we have to have the application to take the game deep.
"If we don't score runs we'll go home, it's as simple as that."
Vettori plans to quit as ODI captain after the World Cup.
New Zealand doesn't have happy recent memories of playing in Bangladesh either, having lost a series 4-0 here last year, but wicketkeeper Brendon McCullum thinks that could work in the team's favour against South Africa.
"It's a real positive for us," McCullum said. "We have some good knowledge about the conditions in Dhaka and it will help us playing against South Africa in those conditions.
"They would probably have wanted some pace and bounce in the wicket, but they won't get that. It is not a bad draw for us at all."
