"The American people are watching this" but expected the vote Friday or Saturday, said Lee, who asked Senate Majority Leader Harry Reid, D-Nev., to not hold the roll call on Thursday.
Reid accused the conservatives of "a big, big stall."
He said he wanted to return the Senate bill to the House as quickly as possible to give GOP leaders there more time to send back an amended bill and avoid a shutdown.
Lee's request also prompted Sen. Bob Corker, R-Tenn., to engage in an icy exchange with Cruz in which Corker accused the two conservatives of seeking a delay because they had emailed their supporters to watch debate on the legislation on Friday.
"My two colleagues, who I respect, have sent out emails around the world and turned this into a show," Corker said, barely masking his disdain. "And that is taking priority over getting legislation back to the House so they can take action before the country's government shuts down."
Just a day earlier, Cruz, a possible 2016 presidential contender, ended a 21-hour speech urging lawmakers to block the Senate bill before Reid amends it to drop the language defunding Obama's 2010 health care law.
That talkathon, and Cruz's strategy, has garnered widespread praise from supporters around the country and become a focus of fundraising appeals by conservative groups.
"We are not going to be complicit in giving Harry Reid the ability to fund Obamacare," Cruz said Thursday.
But Cruz and Lee's effort has earned derision from many GOP colleagues, who see it as a ploy that is doomed and increases the odds of legislative delays that threaten a shutdown. Most Republicans want to avoid a shutdown, fearing blame from voters.
Asked Thursday whether he envisions the House approving a simple Senate-passed bill keeping the government open, House Speaker John Boehner, R-Ohio, told reporters, "I don't see that happening." GOP lawmakers said he signaled the same thing at a closed-door meeting Thursday.