The Scottish Premiership club are trying to persuade Liverpool to allow the 21-year-old Wales international to move north on loan.
Woodburn has made 11 first-team appearances for the Reds and won 10 international caps, along with gaining experience on loan spells with Sheffield United, Oxford and Blackpool.
Neilson said: “We are still negotiating with Liverpool. We are almost done with it, it’s just a case of trying to get things finalised.
“He is still a Liverpool player at the moment and he is one we are really interested in taking here.
“I don’t want to comment too much on it. We are still negotiating, as we are with a few other players as well. We will see how it goes in the next couple of days.
“We have waited a number of months now to get the right players in. We are continuing to do that now with the guys we are trying to recruit. If we need to wait, we need to wait.”