The move comes as GOP senators are increasingly confident they will have the votes to block witnesses from being called.

If Democrats are able to muster four GOP votes to allow witnesses, both sides could then make motions for specific individuals and documents. Under the rules resolution, the Senate would vote on the motions.

But Van Hollen's effort would let Roberts issue subpoenas if he thinks a motion is relevant. The Senate, if it disagreed with his decision, could still overrule him with a simple majority.

"A fair trial includes relevant documents and witnesses. And in a fair trial the judge determines what evidence is admitted," Van Hollen said in a statement. "My motion ensures the Chief Justice will serve the same role as a judge in any trial across our country – to allow the Senate access to the facts they need to get to the truth."