He missed the British & Irish Lions' tour opener against the Barbarians last Saturday but although he was able to train yesterday before the squad moved on to Perth, a subsequent scan has revealed a grade one tear.
Lions head coach Warren Gatland admitted that Kearney is causing 'a little bit of concern' as the tourists prepare for their first match on Australian soil against Western Force on Wednesday.
"We will have to see how Rob responds and potentially if it doesn't heal as quickly as we might like, then we will have to put some plans in place (for a replacement)," explained Gatland earlier today.
"But we want to give him a bit of time. He had a scan in Hong Kong and the medics have put a plan in place for the next week or so.
"We feel like we have enough cover at the moment with Leigh Halfpenny and Stuart Hogg, and Sean Maitland has also played at full-back in the past.
"We will continue to monitor him and will give him time to recover. There is no cut-off point as we believe that we have enough cover at the moment to compensate."
Brian O'Driscoll, who will captain the Lions in Perth, watched the Barbarians game alongside Kearney in the stands and has spoken to his Leinster and Ireland team-mate about his injury.
Admitting that the Louth man has 'felt better and felt worse' in such an injury situation, O'Driscoll said: "It shows what calibre of player he is that he is being given the time (to recover). It also emphasises the impact he had with the Lions four years ago in South Africa."
Like Kearney, 2013 Lions captain Sam Warburton was an absentee for the 59-8 win over the Baa Baas and he was not considered for the Western Force fixture due to a knee injury.
But the Welsh flanker could make his Lions debut against Queensland this coming weekend if his recovery goes to plan, with Gatland stating: "Sam has been doing some of his own training but he had to stay off his knee for five days before he is ready for full contact.
"Hopefully he will train with the other players not in the squad (for the Western Force game) over the next couple of days and put himself in contention for Saturday."