Not all of his Senate Republican colleagues agree.
In interviews with CNN, a number of Republicans want to at least put up a fight in the Senate to show their conservative base that they are trying to strip funding for the organization, which performs abortions and other health-care services to millions of women.
"I have lost many votes in my time," said Sen. Tim Scott, a conservative from South Carolina. "I would rather vote for what I think is right for the American people, making the right decisions, and ultimately if you lose, you lose."
He added: "I think if you lose before you start fighting -- that's not the best or the most desired position from my perspective."
Scott's comments come as congressional Republicans are stuck in a tactical dispute over whether to use a must-pass funding bill to try to gut the organization, which receives more than $500 million annually in government funding. In light of controversial and highly edited videos of Planned Parenthood officials discussing the sale of aborted fetal parts, conservatives have sought to use the Sept. 30 fiscal deadline to force through a government-wide spending bill that eliminates the group's federal funding.
But with President Barack Obama vowing to veto such a measure, and Democrats in the Senate holding enough votes to filibuster it, Senate Republican leaders are worried that such a move would prompt a government shutdown and sharp backlash from voters. In an interview with CNN last week, McConnell said the tactic -- championed by conservative Texas Sen. Ted Cruz -- would be fruitless.
"You could shut down the government and that will not defund Planned Parenthood," McConnell said
. "So that is a strategy that will not lead to a result that I would like. Shutting down the government is something the American people overwhelmingly oppose, and we will not be doing that."
He added that seeking to defund the organization through a must-pass funding bill "wouldn't produce the result that Sen. Cruz and I would both like to produce. So therefore, it would be an exercise in futility."
But few of McConnell's own Senate Republican colleagues are willing to go that far, hoping not to upset the right-flank of their party.
"I think a better way of saying that is: 'It's a difficult challenge but a worthwhile cause," said Sen. Jerry Moran, Republican of Kansas, who is up for re-election next year. "I would not walk away from all options related to funding."
Missouri Sen. Roy Blunt, a member of GOP leadership who also faces voters next year in his conservative state, said it makes sense to try to push a provision in a spending bill to defund Planned Parenthood on the Senate floor, even if there's little chance of success.
"Sometimes, you have to go through an exercise in futility a time or two to truly prove it is an exercise in futility," Blunt said.
But Blunt cautioned: "What I wouldn't want to do is change the topic here from focusing on the conduct of Planned Parenthood to focusing on a shutdown ... If we made a strategic mistake here, it would be changing the topic."
Still, Republican leaders are grappling with how to alleviate demands from their rank-and-file -- particularly in the House. Senior GOP aides say the leadership has yet to settle in on its strategy just yet, so they planned an unusual Wednesday night all-GOP conference meeting to discuss next steps on the funding bill, according to a senior House Republican leadership aide.
The meeting follows a series of "listening sessions" that top House Republican leaders held with rank-and-file members to come up with a legislative strategy around the spending bill and the push to defund Planned Parenthood.
But conservatives are adding pressure to House Speaker John Boehner to take a hardline on the issue. The conservative Republican Study Committee released its wishlist Tuesday, calling for the organization to lose its federal funding. And leaders of the House Freedom Caucus, led by South Carolina Republican Rep. Mick Mulvaney, are circulating a letter urging the GOP to reject any bill that continues funding for Planned Parenthood.
House Majority Leader Kevin McCarthy, R-California, already announced last week that his chamber would vote at the end of this week on a stand-alone bill to bar taxpayer money for Planned Parenthood for a year, while Congress continues investigations. Separately, the House will also vote this week on a bill to impose penalties on medical professionals who fail to provide care for a baby who survives an abortion procedure.
The larger idea is to separate the anti-abortion fight from the spending bill. As part of that effort, Rep. Jason Chaffetz, R-Utah, who chairs the House Oversight and Government Reform Committee, issued a subpoena Wednesday seeking all unedited videos taped by the Center for Medical Progress, the conservative group that secretly taped Planned Parenthood discussions. (Planned Parenthood contends that the videos were doctored and that the group did nothing illegal.)
In the Senate, three GOP-led committees are investigating Planned Parenthood. And McConnell plans to move on a bill to ban abortions after 20 weeks as soon as this week -- a measure that would not be linked to government funding.
The most likely scenario: The House passes a government-spending bill that would also defund Planned Parenthood, which would later fail in the Senate. The Senate would then strip out the Planned Parenthood provision and send it back to the House.
But there are fewer than nine legislative days left before government funding dries up -- and any senator can throw up procedural roadblocks to block the chamber from proceeding. So McConnell and Boehner may have no other choice but to let their chambers vote on a "clean" stop-gap bill that funds the government until December -- and continues funding for Planned Parenthood in the meantime. No final decisions have been made.
"Our members have a lot of ideas on how to proceed and the conference is still discussing them," said Don Stewart, a McConnell spokesman.
McConnell's chief deputy, Sen. John Cornyn of Texas, said in an interview Tuesday that the best idea is to vote separately on anti-abortion bills and take the debate to voters and wait until the next president to battle on the issue. Shutting down the government, he said, would set back the anti-abortion movement.
"This is the way we made progress of the years," Cornyn said. "By having these kinds of votes, differentiating between people who are pro-life and people who are not. And I think we've done a lot to advance the pro-life agenda. So this is not just something we're going to do over the next two weeks."
Still, not everyone agrees.
"I don't think so," said Blunt, the Missouri Republican, when asked if he thought the funding fight could set back the anti-abortion effort. "I don't think we're badly positioned here in terms of what we're doing on the debate of Planned Parenthood and abortion."