After watching his side beat Ulster 20-9 at Welford Road to go top of Heineken Cup Pool Four, Cockerill claimed he was not experienced enough for the England job which became vacant after Johnson revealed earlier this week he would not seek to have his contract extended when it runs out in December.
Johnson was criticised for England's performances on the pitch at the World Cup in New Zealand and also for their behaviour off it.
Asked if he would consider taking over, Cockerill said: "There are better qualified people than me. I have got a lot to learn and it is not an ambition of mine at the present time."
Cockerill described the performance of his players in the win over Ulster as being reminiscent of the Tigers when they were at the pinnacle of the club game in England.
But it was hard earned as the victory was not secured until the final quarter thanks to a try from Matt Smith and a penalty by Toby Flood.
Flood accounted for most of the Tigers' points with his accurate kicking and Ian Humphreys slotted three penalties for the visitors.
Cockerill said: "We did enough to win, our defence and set-piece were very good and I thought we were very consistent over the full 80 minutes.
"It was an old Tigers' performance as we squeezed the life out of them."
But despite going top of the group, Cockerill is not getting carried away as he knows difficult challenges lie ahead.
Leicester now face a tough double header against leading French club Clermont and Cockerill knows both games will test his players more than a resolute Ulster did at Welford Road.
He said: "The group is very open, we are pleased to have two wins but Clermont is a very very hard place to go."
Cockerill will at least have his England players back and firing on all cylinders.
He added: "Manu Tuilagi will be back for that game and Ben Youngs (who came on as a second half replacement against Ulster) looked good tonight as we have looked after him for the past four weeks."
Ulster worked tirelessly but only had three penalties from Humphreys to show for their efforts.
Ulster coach Brian McLaughlin was unhappy his side's performance had not resulted in them at least picking up a losing bonus point.
He said: "We played some fantastic rugby and we are very disappointed that we didn't come away with a bonus point.
"Small things make the difference and we were not careful enough in protecting the ball."
Ulster now have successive games against Italian side Aironi and their coach believes they are still in contention to make further progress.
He added: "We are far from out of it."
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