McConnell said "too much is at stake" to halt an investigation already in progress, though some in his own party appear to leave the door open to such a process.
The Kentucky Republican said from the Senate floor such an investigation into Russian meddling into the US election last year would "only serve to impede the current work being done to not only discover what the Russians may have done but also to let this body and the national security community develop counter measures."
McConnell was referencing the Senate intelligence committee investigation led by Republican Sen. Richard Burr of North Carolina and Democratic Sen. Mark Warner of Virginia.
McConnell added, "Partisan calls should not delay the considerable work of Chairman Burr and Vice Chairman Warner. Too much is at stake."
But other Republican senators refused to rule out the idea of a special prosecutor -- including Sens. Susan Collins of Maine and Lisa Murkowski of Alaska.
"I think it needs to be considered," Murkowski told reporters after McConnell spoke.
Republican Sen. Marco Rubio said that a special prosecutor could later be warranted but wanted the Senate intelligence committee to finish its investigation first.
Meanwhile, Senate Democrats are objecting to committee meetings Wednesday in protest, using Senate rules that require unanimous consent for Senate committees to meet after the Senate has been in session for two hours.
Democrats also continued to hammer President Donald Trump's decision on Comey. Many members of the Senate Democratic caucus showed up for McConnell's opening speech, a rare moment for the US Senate where morning business is typically managed with little fanfare.
"The dismissal of Director Comey establishes a very troubling pattern," Senate Minority Leader Chuck Schumer said on the floor. "This administration has now removed several law enforcement officials in a position to conduct independent investigations of the President and his administration."
Trump fired Comey on Tuesday evening, following recommendations from Attorney General Jeff Sessions and Deputy Attorney General Rod Rosenstein. Democratic lawmakers have called for a special prosecutor to handle an independent investigation into Russian meddling into the US and whether there are any ties between Trump's presidential campaign and Moscow.
Schumer called for a closed and, if necessary, classified briefing for all senators with Sessions and Rosenstein, who could be questioned about Comey's firing.
McConnell said Democrats were "complaining" about Comey's removal after having alleged during the campaign that Comey had taken unfair actions against their party's presidential nominee, Hillary Clinton.
The Kentucky Republican said what the Senate had now was "our Democratic colleagues complaining about the removal of an FBI director whom they themselves repeatedly and sharply criticized."