A hip issue has kept Murray out since Wimbledon and Edmund has emerged as a contender in that time, reaching the semi-finals of the Australian Open earlier this year.
That form will be enough to take top spot in the British game when the latest ATP rankings are released on Monday, but Edmund now wants to prove he can do battle with a fully fit Murray, who is expected to return in June.
"It is humbling to become the British number one," Edmund said, as quoted by The Guardian. "As proud as I am, I would have been much happier had Andy stayed healthy and occupied his place at the very top where he belongs.
"I wish Andy a speedy recovery and I hope to battle it out with him in a more legitimate fashion in years to come."
Edmund has himself been out of action with flu in recent weeks, but he moves up to a career-high 24th as Murray drops to 29th.