Read has not played since breaking the base of his right thumb and undergoing surgery in late April with the fracture healing in a protective cast that was removed on June 10.
The 31-year-old missed the 78-0 victory against Samoa last Friday to ensure he had the best chance of facing the Lions in the three-test series and assistant coach Ian Foster said the team were pleased with his progress.
"We're pretty satisfied," Foster told reporters in Auckland when asked about whether Read would be fit for the opening test at Eden Park on Saturday.
"He has had his first real solid hit out with contact (today) and hopefully he comes through that well. He doesn't want to miss it. He's desperate to play. He's our leader.
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"We have other people who could do a job but to have 'Reado' back would be a real boost for us."
Fullback Ben Smith, who was the team's official vice captain, led the side against Samoa and drew strong praise from coach Steve Hansen for his performance in the role.
Foster said the influence of Read, who took over from long-term skipper Richie McCaw following the 2015 Rugby World Cup, had been important in helping Smith and the team last Friday.
"He's been a big part of the leadership group for a long time," Foster said of Read.
"The way he grew into the role last year, the biggest accolade I can give him is the way he grew people around him.
"He has a very inclusive leadership style, encouraged a lot of decision-making from the playing group around him and earned a lot of respect for that."
(Reporting by Greg Stutchbury; Editing by John O'Brien)
