The Glovers are currently on a winless run of six matches, leaving them two points from safety in the League One table.
Despite doubts over his future at Huish Park, Johnson is remaining optimistic about his side's chances of getting out of trouble in the coming weeks.
"Everyone has a shelf life and if people are deciding that's now, then that's up to them," he told BBC Sport. "From my point of view, we'll do all we can to turn this around.
"My resolve is strong because this is a group I've brought in. There's no excuses that I've taken on somebody else's group. As a coaching team we've got to get them as fit as we can, mentally and physically, and then hope we can get a win pretty soon."
Yeovil are back in action on Saturday when they travel to Chesterfield in the league.