Weekend reports linked him with a switch to the Emirates to replace the inconsistent Wojciech Szczesny, and he has also been linked with Manchester United.
Martin O’Neill: “We wouldn’t want to lose any of our key players next month.
“I haven’t heard of any specific interest, but I certainly wouldn’t want to lose Mignolet or any of the other players who have been so important to us so far this season.
“He’s been outstanding for us so far, so he’s definitely not someone we would want to see leave.
“I haven’t heard of that (the Arsenal link), but I’m not in any position to let our good players go in January. I think we’ve really got to be looking to try to improve our squad next month.
“Once February 1 comes around, it’s a long time after that until the end of the season where you’re not able to change anything.
“In fairness, I think most clubs will be trying to do something. The exceptions are the main sides who have the wherewithal to cope with anything, but numerically, we’re not that strong.
“We have less players than most people in the league, so we really have to look to do something if we can.
"I accept it’s not going to be easy, but some of the fine players we have at the football club probably need a bit of help.
“We’ve managed two wins in the last three games, and I think we’d all admit they were very important matches for us to win.
“We’ve beaten sides in and around the same area, and that’s important. We’re not in any kind of position to start boasting about anything. Absolutely not, particularly with the programme that’s ahead of us.
“But this gives us some confidence to go into the game on Boxing Day.
"John O’Shea will be important against Manchester City.
“I think he felt that last week against Manchester United, his old team, he wanted to play really well.
“Not that he’s got anything more to prove to Man United, he’s had his career there.
“He might have been a bit disappointed, particularly in the first goal we conceded. But he’s made up for that.”