Midfielder Kinsella, who was on Everton's books as a youth player, has been on trial with Micky Mellon's side for the past few weeks.
The 21-year-old was involved in recent friendlies against Stockport and Tranmere and is now targeting a first-team place at Highbury.
"I'm working as hard as I can with the fitness and trying to hopefully secure a place in the starting line-up," Kinsella told the club's official website. "I thought we did very well in the first half (of Saturday's 3-3 draw against Tranmere) and our movement was very quick. We created some good opportunities.
"They started going more direct in the second half and they did create some problems. They were silly goals to give away with two set-pieces and one, big ball down the middle. That's something we need to work on."
Kinsella is hoping those improvements can be made for the Capital One Cup visit of Nottingham Forest next Monday evening, adding: "Forest are going to be a very good side who move the ball quick. The gaffer will be looking at that in training this week."
