Woodward has been touted to take over from Director of Elite Rugby Rob Andrew after England's disappointing RBS 6 Nations campaign.
But Johnson admits his record is not good enough and he would walk away if he felt the team would benefit.
The 40-year-old said: "There's always speculation when results don't come, and that's why there's talk now of Clive taking over Rob's job.
"Clive's got an important job to do at the British Olympic Association.
"My record's clearly not good enough and England should be more successful than that.
"International sport is brutal. If players don't play well they get dropped. If coaches or managers are perceived to be not doing a good job then someone else will come in and do it."