House leaders have agreed to the plan pushed by Lee and Cruz to pass a bill on Friday that keeps the government functioning, but doesn't include any money to implement the law known widely as Obamacare.
Lee said he'll demand a vote on the House plan to show the public where each party stands.
"I do think we will keep all of the Republicans and I hope we might get some of the Democrats to come with us," he said. "Now if they reject it, if Democrats in the Senate reject it, then they've got to come up with a proposal."
Senate Majority Leader Harry Reid blasted the Republican strategy as "absurd" on Thursday and said it was led by Republicans attempting to raise money and grab headlines.
He also called it "a waste of time."
"I want to be absolutely crystal clear. Any bill that defunds Obamacare is dead. Dead," he said.
Sen. Charles Schumer, D-N.Y., said Lee's strategy is based on the faulty premise that Democrats will eventually relent, irreparably damaging President Barack Obama's biggest legislative achievement.
"We will not blink, don't get it into your heads that we will, we won't," he said. "We're unified. We're together. You're not."
Reid also noted the divisions within the Republican Party, where many senior members want to attempt to delay the law, thinking it has a greater chance of success and comes with less political risk.
Sen. Bob Corker, R-Tenn., took to Twitter and wrote: "I didn't go to Harvard or Princeton, but I can count the defunding box canyon is a tactic that will fail and weaken our position."
And some House Republicans have taken shots at Cruz and Lee for saying that they may not have the votes in the Senate, where Democrats control 54 seats, to get the job done.
Cruz said he would be willing to try a talking filibuster, where he would speak on the Senate floor for as long as he is able, to push his position.
"I will do everything necessary and anything possible to defund Obamacare," Cruz said.
Sen. Orrin Hatch, R-Utah, has led repeated attempts to dismantle the health law Republicans despise, but he also disagrees with Lee's strategy saying it likely won't work with President Barack Obama in the White House and Democrats in control of the Senate and if it does lead to a shutdown, Republicans are likely to take the brunt of the blame.
Reid wouldn't say how he would react to the House proposal until after the House votes, but he did say: "They are simply postponing the inevitable choice that they must face, pass a clean bill to fund the government or shut it down."
Utah's three House Republicans have signaled that they will support their party's strategy, while Rep. Jim Matheson, D-Utah, criticized it as a political move that threatened the financial well being of the nation.
The White House insisted Thursday that there is no room for negotiation on this issue, and that Republicans shouldn't be using a government shutdown as a bargaining chip.
If Congress fails to pass a resolution to keep funding the government by Sept. 30, non-essential services will shut down.
"We think it can be prevented," Office of Budget and Management Director Sylvia Mathews Burwell told a group of reporters. "We think there's still time."
Brian Deese, the deputy director of OMB added, "We cannot hold our country, we cannot hold our government hostage to ideological agendas."
Thomas Burr contributed to this report.