The 41-year-old Irishman has masterminded Quins' scintillating start to a campaign that has yielded 12 successive victories in three different competitions.
They are top of the Aviva Premiership by four points, and Saturday's 28-9 Heineken Cup victory over Gloucester at Kingsholm kept Quins on course for a quarter-final place from Pool Six.
Northampton rugby director Jim Mallinder is the bookmakers' England favourite among Premiership contenders, but O'Shea - who previously headed up the Rugby Football Union's academy system - has many admirers.
Johnson quit three days ago on the back of England's dismal World Cup campaign in New Zealand when they made a quarter-final exit to France and were riddled by off-field problems.
O'Shea has no intention, though, of departing a club he has moulded into genuine domestic and European title challengers.
"My job is at Quins, full stop," he said.
"Players like Chris Robshaw and George Robson have committed to 2015, and I want to commit to them.
"We feel we are the start of a journey, not even midway through it.
"I will get a huge amount of pleasure if we are successful and we see a lot of guys go on to represent England.
"I want to be at Harlequins, and I am going to be at Harlequins."
While Quins march on towards potentially pivotal Pool Six home and away encounters against four-time European champions Toulouse next month, Gloucester face making an early exit following two defeats in a row.
First-half tries by full-back Mike Brown and centre Matt Hopper put Quins in charge, but once again it was goalkicker Nick Evans who underpinned victory as he booted three penalties and two conversions.
Quins' England number eight Nick Easter completed an emphatic away success when he touched down eight minutes from time, inflicting a heaviest Heineken home defeat on Gloucester since Stade Francais demolished them 27-0 at Kingsholm six years ago.
Gloucester created their fair share of opportunities in attack - Mike Tindall delivered another strong display following his £25,000 Rugby Football Union fine and being axed from the national squad - yet only had three Freddie Burns penalties to show for it.
And they are now left on the edge of pool stage elimination for a fifth time in their last six European campaigns.
O'Shea added: "I am delighted with that. This is one of the hardest places to come and get a result.
"We looked threatening when we had the ball, and we took our chances. We were clinical. Hats off to the boys.
"We were magnificent in defence. There is an energy among the players, and we have a hell of a lot of heart. Everyone gave everything out there."
Gloucester head coach Bryan Redpath offered no excuses after a defeat that he admitted leaves his team with little realistic hope of reaching the quarter-finals.
"You need a minimum four or five wins in the group stage," Redpath said. "It means we are going to struggle.
"We created enough opportunities, but we didn't take any of them. We didn't look after the ball or the contact area well enough.
"You need to have composure in games like that. If you create, but you don't take, then you are going to get punished.
"We knew it was going to be a tough game and we knew we had to perform, but we didn't. We have got to accept it and deal with it."