By Susan Heavey

WASHINGTON (Reuters) -- Senate Majority Leader Mitch McConnell undercut threats by fellow Republicans to hold the budget process hostage in an effort to defund Planned Parenthood, saying conservatives do not have enough support in Congress to cut off the group, which is under fire from abortion opponents.

Instead, the Senate's top Republican said he is focused on funding the federal government when U.S. lawmakers return next week and will work with President Barack Obama to avert a shutdown, which last paralyzed the government in 2013.

On the possibility of tying measures that kill funding of the women's health group to federal spending bills, McConnell told Kentucky TV station WYMT: "We just don’t have the votes to get the outcome that we'd like.

"The president has made it very clear he's not going to sign any bill that includes defunding of Planned Parenthood, so that’s another issue that awaits a new president hopefully with a different point of view about Planned Parenthood."

Republicans have threatened to push for cutting off the group's federal support, perhaps tying that demand to several must-pass tax-and-spending measures set to come before Congress after its August vacation.

Lawmakers will return to Washington on Sept. 8. Left unresolved, failure to pass those bills could push them to the brink of shutdown by as early as Oct. 1

Congress must pass legislation to fund the U.S. government before the current fiscal year ends Sept. 30, although it could pass a stopgap measure to keep the government operating at current spending levels.

Planned Parenthood has been under fire since the anti-abortion group Center for Medical Progress began posting secretly recorded videos online. The center says the tapes show Planned Parenthood engaged in illegal sales of fetal tissue. Planned Parenthood counters that the videos were distorted and says that it did nothing wrong.

The center on Tuesday released a related video and, in a letter dated Aug. 31 to McConnell and other top lawmakers, urged Congress to continue investigating the group.

Later on Tuesday, Planned Parenthood and the Women's Med Group Professional Corporation jointly filed a federal complaint to block efforts to shut down two women's health clinics in Southwest Ohio. The complaint follows similar suits Planned Parenthood has filed in recent weeks in Florida, Louisiana and Alabama.

U.S. health officials have warned states that efforts to shut down Planned Parenthood clinics may be a violation of federal law.

(Reporting by Susan Heavey; Additional reporting by Megan Cassella; Editing by Kevin Drawbaugh, W Simon and Leslie Adler)