Lions coach Sir Clive Woodward plans on O'Driscoll leading the Lions out at Rotorua for the opening tour game against Bay of Plenty.
Welsh number eight Michael Owen will lead the side against Argentina at the Millennium Stadium, with Jonny Wilkinson as his vice captain.
"This is the Lions tour of New Zealand and I think it is important that Brian leads the team out in that game," said Woodward.
"I spoke to Brian about the captaincy. He's clearly not going to be playing every single game. There will be other players given the honour of captaining the Lions.
"We were unanimous in wanting Michael to be captain in the opening game. If all goes to plan, Brian will be captain of the Lions for the opening game of the New Zealand tour."
The Lions side for the historic home Test includes six Irishmen. Geordan Murphy starts at fullback with Denis Hickie on the wing and Gordon D'Arcy in the centre.
Prop John Hayes and hooker Shane Byrne pack down in the front row with Malcolm O'Kelly starting at lock alongside England's Danny Grewcock.
Woodward did not consider Gavin Henson, Tom Shanklin, Richard Hill, Matt Stevens or Will Greenwood for selection as all are carrying minor injuries.
"I just don't want to take any chances in the first game. All could have played on Monday night, they are minor injuries.
Given we have almost two weeks before the first game in New Zealand this seems the wise thing to do," said Woodward.
The Lions are also without Sale quintet Jason Robinson, Charlie Hodgson, Mark Cueto, Andy Sheridan and Andy Titterrell, who are preparing for the European Challenge Cup final.
France-based pair Gareth Thomas and Stephen Jones are both on club duty - with Thomas preparing with Toulouse for Sunday's European Cup final - but should be available to face Bay of Plenty.
"I have told the players they will definitely start one of the early games," said Woodward.
"Eleven players were training this afternoon (Wednesday). I would expect the majority of those to be involved in the next game. That is how we will try and work it, but there will be a crossover."