But even as Spicer spoke, TheHill.com reported that the number of Republican House members who have stated they will oppose the bill is 22. That's the maximum number of defections the GOP can have, given the fact that all Democrats in the House are expected to vote against it.
And because there are several dozen more GOP members who are undecided, it is very unlikely that Republican leaders could win passage if they called the vote Monday.
Earlier Monday, National Economic Council Director Gary Cohn, during an appearance on CBS, predicted the bill would pass this week.
"This is going to be a great week," Cohn said. "We're going to get health care down to the floor of the House. We're convinced that we've got the votes."
When Spicer was asked later if Cohn had been overly optimistic, the press secretary said that "I would never get in front of the speaker," referring to House Speaker Paul Ryan, R-Wis. Spicer noted that Ryan, along with the rest of the House leadership, "ultimately" will decide when the right time is to call for a vote.
In late March, that leadership, along with Trump, were badly embarrassed when they canceled a planned vote on the replacement bill after it became clear it would fail to win passage
Since then, the bill has undergone several changes, each of which has been followed by predictions that passage was close, followed by walking back of those projections by the White House, or Republicans in Congress, or by both.
That is despite the fact that Republicans, who for years campaigned on promises of fully repealing Obamacare, control both the White House and the two chambers of Congress.
Last week, the conservative House Freedom Caucus said it would back the bill after changes were made to allow states to get waivers that would allow insurers not to cover certain minimum health benefits, and also to allow insurers to charge people with health issues higher premiums.
While some conservatives might have been persuaded to vote for the bill by those changes, a number of moderate Republicans either continue to have problems with the bill, or were turned off by the changes themselves.
The Republican holdouts are concerned that the bill would lead to steep increases in both the number of uninsured people, and premiums for millions of Americans who continue to buy private individual insurance plans. They also fear that they will be blamed for those effects during the 2018 congressional elections.
Watch: Time to bet on insurers