Step 1: Identify the provision.
The question is about transferring a pending civil case from one district to another district within the same state. The Code of Civil Procedure, 1908 covers this under Section 24.
Step 2: What Section 24 says.
Section 24(1) gives the High Court and the District Court power to transfer any suit, appeal, or proceeding to a court subordinate to it and competent to deal with it, at any stage.
Step 3: Focus on inter-district transfer.
When the case has to move from one district to another district within the state, the District Court of one district cannot reach into another district. The court that sits above both districts is the High Court. So the High Court orders such a transfer.
Step 4: Apply to the facts.
Since the request is to shift the case across districts in the same state, the High Court has the power to order it.
Step 5: Remove the wrong options.
It does not need agreement of both parties, it need not wait until the trial is over, and the court where the suit is pending cannot transfer the case to another district on its own. So those options are wrong.
Step 6: Conclude.
The inter-district transfer may be ordered by the High Court.
\[ \boxed{\text{By the High Court.}} \]