The Leinster full-back damaged his bicep towards the end of Saturday's 63-10 win over Italy in Rome.
The Irish Independent has reported that the injury could require surgery and Kearney is set to miss this week's three-day training camp in Monaghan.
Ireland have declined to comment on the extent of the injury but will issue an update on the situation on Friday.
The 30-year-old is in a race to be fit for the matches against France, Wales and England.
Simon Zebo is the most likely candidate to assume the number 15 jersey should Kearney be missing, with Ulster's fit-again Andrew Trimble coming on to the wing.
Robbie Henshaw is set to train this week despite suffering a dead leg.
The centre was forced to retire hurt with the injury against the Italians but is likely to join the Irish squad when they get together on Wednesday.
Flanker Peter O'Mahony, who was not considered for Ireland's opening two Six Nations games because of a hamstring injury, trained with Munster on Monday, but will not play for the province against the Ospreys on Saturday.
Meanwhile, Jamie Heaslip has agreed a new contract with the Irish Rugby Football Union, which will see him continue to play for Leinster and Ireland until the end of the 2019 Rugby World Cup in Japan.
The 33-year-old number eight has won 93 caps for Ireland, scoring 13 tries and has started 36 of Ireland's last 41 Test matches.

Benzema is being investigated for his part in an alleged plot to blackmail France team-mate Mathieu Valbuena.
The Real Madrid player - who denies wrongdoing - was not allowed to contact Valbuena, meaning they could not be in the same squad, but an appeals court has partially lifted some restrictions.
However Benzema, who is injured, was suspended from France duty in December.
That ban was due to stand until the end of court proceedings.
An investigating judge last month lifted the order keeping Benzema and Lyon playmaker Valbuena apart, but the prosecutor appealed. That was dismissed on Friday.
Benzema's lawyer, Sylvain Cormier, described the decision as "a first step".
The 28-year-old forward faces preliminary charges of conspiracy to blackmail relating to an extortion scam over a sex tape involving Valbuena.
If found guilty, he could face a minimum of five years in prison.
According to French media, Benzema mentioned a sex tape to Valbuena, 31, during a national team training session at Clairefontaine on 5 October.
Benzema told police he was advising a friend about what to do in a difficult situation.
But he could face charges for complicity in attempted blackmail and participation in a criminal conspiracy.
Rob Kearney could be ruled out of the remainder of Ireland's Six Nations campaign with an arm injury.
[NEXT_CONCEPT]
Striker Karim Benzema will be eligible to play for France at Euro 2016 after some legal restrictions were lifted.