Savage, who left Brighton at the turn of the year after contract talks broke down, played in a 1-0 reserve defeat by Rangers on Tuesday.
Kilmarnock assistant boss Billy Brown now wants to take a further look at the 26-year-old.
Brown said: "He wasn't 100% fit, he has not played since Boxing Day.
"He has not been training with a team so he has lost a wee bit of his match fitness and he looks the type that has got to be sharp.
"We have invited him back up to train so we will just have to wait to see if he wants to do that."
Savage's agent, Kevin Horton, said his client was "highly likely" to take up that offer following his initial spell training with Killie.
Horton added: "He thoroughly enjoyed it and was impressed with both the club and Jim Jefferies in particular, and also the lads."