Sen. Mike Lee dismissed the odds of a government shutdown Sunday morning, downplaying the stakes of the ongoing showdown between Republicans and Democrats over Obamacare and keeping the government running.

When asked by NBC's David Gregory on "Meet the Press" whether there would be a shutdown when the government's funding authority expires at the end of the month, the Utah Republican simply responded "no."

"We all know that the government's going to be funded," Lee said. "The question is whether it will be funded with Obamacare or without."

Along with Sen. Ted Cruz of Texas and other conservatives, Lee has been pushing for Republicans to pass a spending bill that would fund the government but not the president's health care law. The GOP-run House of Representatives did just that Friday, and Lee insisted Sunday that he will have support for the House's measure in the Senate, in which Democrats hold the majority.

Lee said that "we've seen that the American people are rallying around this idea" and that "we have support in the Senate."

He said he was counting on all 46 Senate Republicans backing the defund-Obamacare push and expressed hope that "a few Senate Democrats, particularly those from red states who are up for re-election this year, will consider joining us."

Lee's chief ally in the movement to remove funding for Obamacare, Cruz, also expressed optimism about their efforts. Cruz outlined a complicated path forward for advocates of defunding Obamacare, one that involves GOP senators blocking a procedural vote on the House bill in order to prevent Senate Majority Leader Harry Reid from stripping out the provision that would deny funding for Obamacare. Cruz also suggested that, in the case that Reid blocks the entirety of the House measure, Republicans in the House could begin passing "smaller continuing resolutions" that fund the government, but not Obamacare, piece-by-piece.