Kidney left his post on Tuesday after the Irish Rugby Football Union decided not to offer him a new deal when his current contract expires on June 30. O'Shea, the former Ireland full-back, and Saracens' boss Mark McCall were immediately tipped for the job but both remain committed to their clubs.
When asked about the prospects of him taking over as Ireland coach, O'Shea said: "Absolutely not. I am contracted to Harlequins until the end of the 2013-14 season. I will be honouring that commitment and, hopefully, continuing beyond it."
He added: "Of course with the fixtures this weekend there are lots of questions but I've made my position very clear."
O'Shea is preparing Harlequins for a Heineken Cup quarter-final battle with Munster on Sunday, 24 hours after Saracens tackle Ulster at Twickenham.
Saracens chief executive Edward Griffiths said McCall is "totally committed" to the club after signing a new two-year deal and that there had been no contact from the IRFU.
Ewen McKenzie, the former Australia prop, has been linked with the Ireland job since announcing his intention to step down as Queensland Reds head coach at the end of the current Super Rugby season.
McKenzie is the favourite in the eyes of the bookmakers, who are also offering short odds on the former Wales head coach Mike Ruddock and Leinster coach Joe Schmidt.
Les Kiss, Ireland's defence coach under Kidney, will step up to be interim head coach for the summer Tests against the United States in Houston on June 8 and Canada in Toronto a week later.
Kidney led Ireland to the Grand Slam in 2008 but he departed after a poor Six Nations campaign. Ireland lost to England, Scotland and Italy to finish fifth and drop to their lowest ever world ranking of ninth.