The special counsel's office declined to comment for this article.
Representatives for the top Senate and House Democrats, Chuck Schumer and Nancy Pelosi, did not immediately respond to requests about whether the leaders agree with Coons' view.
Trump's personal legal team has partly cited the midterms as it pushes for a swift conclusion to the Mueller investigation. Rudy Giuliani, one of the president's lawyers in the Russia probe, told CNBC on Wednesday that he would want a potential Trump interview with Mueller's team to take place "well before midterms so they can wrap up [the] case in September."
Trump's fixation on the investigation has helped to keep it in the public eye. He tweeted frequently in recent days about the Russia investigation and unfounded accusations that the Obama administration planted a spy within his campaign.
"Why didn't President Obama do something about the so-called Russian Meddling when he was told about it by the FBI before the election?" he asked in one tweet on Sunday.
Republicans have tried to focus their messaging on a strong economy and their law cutting tax rates for corporations and most individuals ahead of the midterms. On Tuesday, Trump appeared to acknowledge that he should spend more time talking about policy rather than the Russia investigation.
"Sorry, I've got to start focusing my energy on North Korea Nuclear, bad Trade Deals, VA Choice, the Economy, rebuilding the Military, and so much more, and not on the Rigged Russia Witch Hunt that should be investigating Clinton/Russia/FBI/Justice/Obama/Comey/Lynch etc.," he wrote on Twitter.
— CNBC's Brian Schwartz contributed to this report