Smith suffered the injury during Northampton's European Cup quarter-final defeat to Toulouse on Friday night and is expected to miss the remainder of the Saints' season.
There is still time for the 33-year-old to regain fitness before the Lions leave for New Zealand, after taking on Argentina in Cardiff on May 23.
But if Smith, who has started the last six Lions Tests against South Africa and Australia, is to make a third tour then Woodward will have to name yet another injured player in his squad.
Jonny Wilkinson, Richard Hill, Colin Charvis, Phil Vickery and Will Greenwood are all selection candidates who missed the RBS 6 Nations through injury and are at varying stages of their rehabilitation.
Fortunately for Woodward, his loose-head options are well-stocked with Wales' Gethin Jenkins and England's Graham Rowntree both in prime form.
But it is on the other side of the front row that Woodward could be facing difficulties.
England tight-head prop Julian White suffered a neck injury against Wales on the first weekend of the Six Nations and has not played since.
White had been pencilled in to play for Leicester against Leinster at the weekend but suffered a setback in a live scrummaging session last week.
Vickery, another leading contender for the number three jersey, suffered a broken arm in February that has still not healed properly and he faces another month on the sidelines.