By Manu Raju and Jeremy Herb, CNN

(CNN) -- A growing number of key Republicans are sending this message to the leaders of the congressional committees investigating potential Trump campaign collusion with the Russians: Wrap it up soon.

In the House and Senate, several Republicans who sit on key committees are starting to grumble that the investigations have spanned the better part of the past nine months, contending that the Democratic push to extend the investigation well into next year could amount to a fishing expedition. The concerns are in line with ones raised by President Donald Trump, who has publicly and privately insisted he's the subject of a "witch hunt" on Capitol Hill and by special counsel Robert Mueller.

Democrats, meanwhile, are raising their own concerns that the congressional Russia probes are rushing witnesses -- including the testimony of President Donald Trump's son-in-law Jared Kushner -- as well as stalling appearances of other key Trump associates.

CNN interviewed more than two dozen lawmakers and aides on the three committees probing Russia's election meddling and possible collusion with Trump's team, which highlighted the partisan tensions and suspicions bubbling beneath the surface -- and increasingly out in the open.

Sen. Jim Risch, a senior GOP member of the Senate intelligence committee, said "nobody wants to move this so quickly that we miss something," but added: "The question is how many weak leads can you follow?"

"We're a long ways down the line," Risch, an Idaho Republican, told CNN. "And with any of these things, the law of diminishing returns comes to play, and that's where we are right now, by any description."

"I don't see any reason why it couldn't be done this year," said Sen. John Cornyn, the No. 2 Republican who sits on the intelligence and judiciary committees, calling for a final report in time to make changes ahead of the 2018 elections to prevent against more Russian cyberattacks.

The comments were echoed among influential Republicans across the three panels investigating potential Russian collusion with the Trump campaign. And the remarks present a fresh challenge for the GOP leaders of those committees, who are trying to navigate pressures from their members to finalize the inquiries while also attempting to chase down all relevant leads, which take time to pursue.

Rep. Mike Conaway, the Republican who is leading the House intelligence committee's Russia investigation, said this when asked about the timeline for issuing a final report: "Absolutely sooner than later. As soon as we get the things done we need to do in order to get the report written and finalized, we'll do that."

Conaway declined to put a date on a final report, however.

Sen. Richard Burr, the chairman of the Senate intelligence committee, said Tuesday it's still his "aspirational goal" to finish the investigation this year.

"You can't fabricate people to talk to," the North Carolina Republican said, adding there's a "finite number of folks involved" in the Russia matter, saying the panel "will exhaust" the interview process after speaking to everyone they think is relevant.

"I just can't tell you whether that's December 15 or January 30," he said.

The comments come as the top Democrat on the House intelligence committee, Rep. Adam Schiff, raised fresh concerns in a recent Washington Post op-ed about GOP efforts to "halt the congressional investigations rather than allow the evidence to dictate the pace and breadth of our inquiry." He said that some witnesses were being "rushed" before Congress before the committee had received the relevant documents, and other key witnesses have yet to be called before the panel.

In interviews with CNN, several Republicans on the committee sharply pushed back at Schiff's contention.

"It's nuts. It's just not true," said Rep. Chris Stewart, a Utah Republican who sits on the House panel, adding that it's "no question" Democrats were eager to extend the investigation into the midterm election season next year.

"For sure they are," Stewart said. "There's no question, their incentive is to keep this narrative alive for as long as they can and as closer to the 2018 election as they can."

Rep. Mike Turner, who also sits on the House panel, took a shot at Schiff.

"For a guy who continues to try to make his mark on TV on this topic, I'm certain that he will be very sad to see any portion of the investigation come to a conclusion, even if they're timely and accurate," Turner, a Republican from Ohio, told CNN.

In response, a Democratic committee aide said that Democrats on the panel have requested "dozens of witnesses who have not yet been scheduled," saying that the "unwillingness to bring them in raises questions about whether the majority is committed to being thorough and truly following the evidence where it leads." The aide added Kushner was questioned earlier this summer by the panel before the Trump campaign provided documents to Congress. Democrats have called publicly for Kushner to return to the Hill to answer more questions.

"This is an incredibly complex investigation that is global in scale, and will take time," the aide said, adding that pressure from Trump and his allies like Steve Bannon should not "undermine or impede our work."

The House panel has been the source of an enormous amount of tension for months -- something that was readily apparent last week when former President Barack Obama's ambassador to the United Nations, Samantha Power, testified about questions Republicans had about whether she improperly identified individuals in classified intelligence reports.

After the session, which lasted more than three hours, Democratic Rep. Eric Swalwell said Republicans were far more interested in questioning Power than key players in the Russia investigation, including Kushner. He said the GOP-led panel keeps bringing "irrelevant witnesses" in to testify.

"This interview lasted longer than almost any interview of a Trump campaign or Trump family member," Swalwell asserted. "And I think that says a lot about the direction of this investigation."

Democrats in the Senate, too, are warning Republicans not to pull the plug too early.

"I think it is exceptionally important that important inquiries like this be driven by the facts and the evidence, rather than just plucking a date out of the air and saying that's that," said Sen. Ron Wyden of Oregon, a Democrat who sits on the Senate intelligence panel.

Sen. Mark Warner, vice chairman of the panel, said the investigation "is going to take as long as it takes to get all the witnesses and all the facts," adding that "new strands" need to be fully investigated.

"We've seen an awful lot of fish already," said Democratic Sen. Martin Heinrich when asked about criticism of mounting a fishing expedition.

At the same time, a partisan fight is brewing on the Senate judiciary committee where some Democrats are growing impatient at the inability of the chairman of the panel, Chuck Grassley, and the top Democrat, Dianne Feinstein, to reach a deal on the next steps of the committee's investigation.

"It's clearly the other side," Feinstein said when asked about the failure to get an agreement, declining to discuss the matter further.

"There hasn't been progress in terms of witnesses coming before us," added Senate Minority Whip Dick Durbin, a Democrat who sits on the committee. "I think there's a lot of frustration."

Grassley referred questions on the matter to a spokesperson, who declined to discuss the private negotiations. But he did agree that a public hearing to hear testimony from Trump's eldest son -- Donald Trump Jr. -- was still "unavoidable," something he asserted earlier this month.

Still, even on Grassley's committee, some GOP members say their panel should steer clear of the Russia matter altogether, leaving the issue to the intelligence committees instead.

North Carolina Sen. Thom Tillis, a GOP member of the Senate judiciary committee, described it this way: "I think having fewer shiny objects coming out of Congress ... will help the integrity of the process."