United are top of the Women’s Super League and three points ahead of third-placed Arsenal.
Russo, 23, has scored five times in nine WSL appearances this season and is free to talk to other clubs outside of England as she has fewer than six months remaining on her contract.
Asked if he had spoken to Russo since Tuesday’s deadline, Skinner said: “I spoke with Alessia this morning and we had a really good meeting, really positive.
“She has a great attitude and is exactly the type of person we are looking to keep here, keep having negotiations with and keep conversations around that.
“She wants to win here. I can’t speak any more highly of her attitude. She just gets on with her job. We had a good moment this morning, had a good meeting and she trained really well this morning.”
Skinner also said it was “huge” for United’s title challenge to keep Russo for the remainder of the season.
“What’s clear from this window is that as Manchester United we don’t want to be a selling club,” he added.
“We want to be a team that builds to try and push for titles. We’re a club that has fierce ambition.”