The England prop suffered a knee ligament strain during Bath's Zurich Premiership victory against London Irish 18 days ago.
It initially cast doubt over the 22-year-old's chances of touring New Zealand with the Lions this summer, giving coach Sir Clive Woodward cause for concern in a front-row department where injuries have proved a recurring theme.
Leicester prop Graham Rowntree is currently battling back to fitness following a knee injury, although he could be involved in Saturday's Zurich Premiership final against Wasps, while Gloucester's Phil Vickery (broken arm) looks unlikely to win his race for a seat on the plane.
But Stevens, one of England's success stories during a disappointing 2005 Six Nations campaign, believes he will be fit in time for departure on May 25.
"The rehabilitation is progressing well, and I will start running again (on Thursday)," he said.
"The knee is feeling good, and I am remaining extremely positive about it all. I will have fitness tests, but it looks as though I will be on that plane and also available for the Argentina game (May 23) if required.
"Lions selection is like a dream come true. I am really looking forward to getting out there and being part of what is going to be an awesome squad of players and coaches."
In the meantime, Stevens has sorted out domestic matters by agreeing new terms at the Recreation Ground.
"There is a great atmosphere here, a strong playing and coaching set-up, and I feel that I am in the right environment to continue to develop my game," he added.
"All the signs are good for next season in the sense that we are well-positioned to build on what we have achieved over the past couple of seasons."