A hamstring injury has restricted the Celtic defender to just two league games since December but he came through 90 minutes against Qatar on Friday night unscathed.
He told the Scottish Football Association’s website: “Of course, everybody wants to play in it. It is a big squad and we are all fighting for places and I think that brings out the best in everybody.
“You saw against Qatar everybody fighting for their place so I’m desperate to be involved next week as much as anybody else who was out there.”
Several players were rested for the Easter Road encounter and Steven Whittaker, Russell Martin, Andy Robertson, James Morrison and Steven Fletcher could all return to the starting XI in Dublin next weekend.
Speaking after the 1-0 win over Qatar, Mulgrew said: “It is always good to win and also to get a clean sheet. It is a positive result for us. There is a lot we could have done better, we know that as a squad.
“But we are happy with some of the play. We created chances and didn’t give much away at the back so we are happy with that and now we move on. The game in Ireland is a whole different game and we will prepare for that this week.”
You can follow all the action from Dublin as Republic of Ireland face Scotland on June 13 live on Sky Sports 5HD and Sky Sports 1HD from 16:00.