LA Galaxy forward Keane enjoyed a successful loan spell at Aston Villa at a similar stage last term, scoring three goals in seven games for the Midlands club.
Republic boss Giovanni Trapattoni has revealed Keane is looking to fix himself with a club on a short-term basis to ensure he is fully fit for his country's crucial World Cup qualifiers against Sweden and Austria in March.
"Robbie spoke to me about this situation after last month's game against the Faroes Islands," said Trapattoni.
"If he has this opportunity, it's his decision. He has to make up his own mind, but in life it's very important to have new challenges and if it suits him, it'll be an injection for him.
"If he stays in America, he would have no football for three months, but his club have a tour in Malaysia.
"He has family - a wife and child - and until now he has not decided about what to do."