Atkins is aiming for the July deadline so that an appointment coincides with its preliminary year-end results.
A source close to Atkins said it might be difficult to prise the chosen candidate from their post. The source said: "At the end of the day there is no guarantee that someone will leave a perfectly good job."
Jeffries has combined the chief executive and chairman roles since the departure of chief executive Robin Southwell in October.
Atkins did select a successor to Southwell earlier this year but the candidate turned down the post.