The 31-year-old striker was released by Ipswich in January and has scored four goals in as many games for Alex's second string since joining Davis' squad at Reaseheath.
Ellington told BBC Radio Stoke: "It would be nice to get something sorted, 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.
"But, looking from the outside, coming from the Championship to League One, it does not look like there is much difference and you never lose the hunger to score goals."
Davis maintains he is still pondering the merits of adding the well-travelled forward to his squad.
"Obviously I have to think about the future of the club and that is our young players and I don't want to do anything which is detrimental to them," he told Crewe's official website.
A player of Nathan's quality is difficult to bring into a football club like ours, he is a proven goal-scorer at the very top level and financially it is difficult. If we are to do something then we'd have to see what sort of financial packet he'd be looking for, whether we could afford it, and whether he wants to do it.
"I haven't totally made my mind up. I can only judge by what I see and he has caught the eye, but it is a difficult decision because of the impact it could have on our young players. But it could be positive because they could learn from him. It's whether the positives outweigh the negatives."