The top Republican crafting the Senate’s sweeping immigration-reform legislation acknowledged Sunday the bill still has flaws, while a fellow GOP senator said their party blocking its passage will only add to their “demographic death spiral.”

Florida Republican Sen. Marco Rubio, the son of Cuban immigrants and potential 2016 presidential candidate, said roughly 95 percent of the bill is in “perfect shape” and that the full chamber debates are off to a good start.

However, he expressed concerns about whether the legislation ensures adequate border security and said Americans have “valid issues” about the matter.

"The vast majority of Americans, the vast majority of conservative Republicans, are prepared to support immigration reform, but only if we can ensure that we're not going to have another wave of illegal immigration in the future," he told ABC’s “This Week."

He declined to say whether he would vote for the measure unless changes are made.

South Carolina Republican Sen. Lindsey Graham, who also helped write the bipartisan immigration bill, told NBC’s “Meet the Press” that conservatives trying to block the measure will doom the party and all but guarantee a Democrat will remain in the White House after the 2016 elections.

"After eight years of President Obama's economic policies, and quite frankly foreign policy, people are going to be looking around," he said. "But if we don't pass immigration reform, if we don't get it off the table in a reasonable, practical way, it doesn't matter who you run in 2016. We're in a demographic death spiral as a party and the only way we can get back in good graces with the Hispanic community in my view is pass comprehensive immigration reform. If you don't do that, it really doesn't matter who we run."

A Democrat also involved in crafting the bill, New Jersey Sen. Robert Menendez, went a step further by predicting on CNN’s “State of the Union” that "there'll never be a road to the White House for the Republican Party" if immigration overhaul fails to pass.

The Senate last week overcame a procedural hurdle in moving forward on the first immigration overhaul in a generation.

Lawmakers from both parties' voted to begin formal debate on a proposal that would give an estimated 11 million immigrants in the U.S. illegally a long and difficult path to citizenship.

The legislation also creates a low-skilled guest-worker program, expands the number of visas available for high-tech workers and de-emphasizes family ties in the system for legal immigration that has been in place for decades. It also sets border security goals that the government must meet before immigrants living in the U.S. illegally are granted any change in status.

In addition to tougher border-security measures, Republicans are demanding stricter standards for who qualifies for government programs such as Social Security and health care.

Rubio is trying to balance concerns from his party's conservative flank that has great sway in picking a nominee with the political attempt to win over Hispanic and Asian American voters who overwhelmingly favored President Obama's re-election in 2012.

Further complicating Rubio's presidential aspirations, the Republican-led House is considering its own version of immigration proposals that more closely follow their own perspective, which hews toward Tea Party principles.

In 2012, Obama won re-election with the backing of 71 percent of Hispanic voters and 73 percent of Asian voters. A thwarted immigration overhaul could send those voting blocs more solidly to Democrats' side in future elections. That has led some Republican lawmakers to support immigration reform, but the party's conservative base still opposes any legislation that would create a pathway to citizenship for immigrants living here illegally.

Democrats are well aware of the numbers.

"I would tell my Republican colleagues, both in the House and the Senate, that the road to the White House comes through a road with a pathway to legalization," Menendez said. "Without it, there'll never be a road to the White House for the Republican Party."

The Associated Press contributed to this report.