Wigan chairman Dave Whelan claims his club have been granted permission to speak to Brentford boss Uwe Rosler.
Stevan Jovetic's agent insists that the Manchester City striker has settled into life in England, despite exit rumours.
The 74-year-old Italian went into the double-header against Sweden and Austria amid claims that defeat in Stockholm could not only end his side's hopes of reaching the 2014 World Cup finals in Brazil, but could also signal the end of his reign.
Former Reading manager Brian McDermott has been linked as a potential successor.
In the event, and despite a controversial late reshuffle in his team selection, Trapattoni returned from Sweden with a hard-earned point and is preparing for Tuesday night's clash with Austria, in which victory would catapult Ireland into second place in Group C.
The manager has grown accustomed to the dissenting noises off-stage and has allowed them to wash over him, and O'Shea is equally adamant that they do not affect the players either.
"Not in terms of in our minds," he said. "It's very clear for us: he is the manager who is trying to get us to the World Cup.
"It's fairly straightforward, to be honest."
What is also straightforward is that the Republic's creditable performance and result against the Swedes on Friday will count for little if they do not make the most of home advantage to claim the three points when Austria visit the Aviva Stadium.
Austria leapfrogged the Republic with a 6-0 demolition of the Faroe Islands in Vienna on Friday evening, although their advantage is only goal difference.
But with Sweden not in action this week, a win would put Trapattoni's men second in the Group C table behind Germany.
O'Shea said: "There's a good positive atmosphere, but also we are fully aware of the dangers we face against Austria.
"The job is only half-done. Four points, I suppose, was the target before the two games and if we can do that after the game on Tuesday night, we will be very pleased.
"But believe me, we are not taking Austria lightly. They have shown it in the group beforehand.
"We were watching the clips against Germany, and Austria caused Germany lots of problems. They will be confident from scoring six goals the other night as well.
"But we will have grown in confidence too. Some of our younger players will have grown a lot in confidence from the other night, so it's a game to look forward to."
The game is one of three successive qualifiers on home turf for Ireland, and while their recent record in Dublin has been patchy, O'Shea knows now is the time to recreate the kind of fortress Ireland once enjoyed at the old Lansdowne Road.
Asked if the balance of the group had changed as a result of what happened in Stockholm, the 31-year-old said: "If we can beat Austria, it will do.
"We have three home games in a row now, so it's really back in our court if we can take advantage of that.
"The confidence should have grown from the other night and we can look forward to a big game at the Aviva, but also a tough game as well."
The Republic will have to pose a greater threat than they did in Sweden, where they did not manage a shot on target.
West Brom striker Shane Long had the best of their opportunities, but fired over, reminding team-mate O'Shea of his days in Gaelic football.
He said with a smile: "Longy was unlucky that one time. He sent the defender on to the ground and the ball just hopped up, unfortunately.
"He connected with it beautifully, but he must have thought he was back playing for Tipperary again.
"Fingers crossed if the manager does select him to start again, he will cause Austria lots of problems. We will have to wait and see."