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.
Rob Kearney could be ruled out of the remainder of Ireland's Six Nations campaign with an arm injury.