The 31-year-old has been on trial with the League One outfit for the past month after being released by Ipswich Town at the start of February.
He has played three Under-21 league games for Crewe and scored in each of them.
"It would be nice to get something sorted," Ellington told BBC Radio Stoke. "But I'm just happy I've had the chance to come here and train and keep fit.
"It's good for both parties at the moment. I didn't get to play too many games at Ipswich.
"Crewe have always been a good footballing side. They're a club who do things the right way. A lot of players have moved on from here to bigger and better things."
Ellington has scored 144 goals in 329 career appearances since making his debut for Bristol Rovers in 1999.