With the Senate set to begin full debate this week on a bipartisan bill to give Congress the authority to review and potentially reject any Iran nuclear deal, some members are being warned about pushing amendments that change the bipartisan legislation and nix its support.

"Anybody who monkeys with this bill is going to run into a buzz saw," Sen. Lindsey Graham, R-S.C., warned ahead of the floor debate.

And Sen. Bob Menendez, D-N.J., is urging fellow senators to stick with the plan that recently emerged from the Senate Foreign Relations Committee.

The high-profile debate comes as negotiators from the United States and five other countries try to reach a final deal with Iran by June 30 to curb the country's nuclear-enrichment program, in exchange for relief from sanctions choking its economy.

The parties will meet again this week on the sidelines of a U.N. conference in New York.

The Senate bill was approved 19-0 by the Foreign Relations committee and has 62 co-sponsors from both parties. It is expected pass the GOP-led Senate and House.

Some lawmakers, however, want changes that risk them losing the support of President Obama, who grudgingly backed the measure, and his fellow Democrats.

If there is a final deal with Iran, Obama can use his executive authority to ease some sanctions on his own and work with the European Union and the United Nations to lift others. Obama also can waive sanctions that Congress has imposed on Iran, but he cannot formally lift them.

The bill would block Obama from waiving congressional sanctions for at least 30 days while lawmakers weigh in.

If 60 senators vote to disapprove of the deal, Obama would lose his waiver power altogether. The president is betting he will not.

If Congress disapproves, the president will almost certainly respond with a veto. As long as he can get more than one-third of the Senate to side with him, he can prevent his veto from being overridden.

Backers of the bill are trying to keep lawmakers focused on how it would give Congress a say on a critical national security issue. They say the measure is not meant to be about how Iran increasingly is wielding influence in the Middle East, its support of terrorist groups or human rights violations. They worry that adding too many divisive amendments would cause Democrats to drop their support.

Even so, some senators are proposing amendments to pressure Iran to end its support of such groups, stop threatening to destroy Israel and recognize its right to exist, and release U.S. citizens held in Iran.

Other amendments would prevent sanctions relief if Iran cooperates with nuclear-armed North Korea or until international nuclear inspectors are guaranteed access to Iranian military sites.

GOP Sen. Ted Cruz of Texas, a presidential candidate, has an amendment with Sen. Pat Toomey, R-Pa., that would require Congress to sign off on any final nuclear deal, not just disapprove of it. An amendment from Sen. Ron Johnson, R-Wis., would make any deal a treaty, thus needing to be ratified by two-thirds of the Senate.

"The president should have to get 67 votes for a major nuclear arms agreement with an outlaw regime," said Sen. Tom Cotton, R-Ark.

Cotton wants to lower the number of votes needed to reject a deal from 60 to 51. That means opponents of any deal would only need Republican votes to sink it.

He also wants to see amendments requiring that Congress be notified of any violations of an agreement, not just ones that are legally defined as material breaches.

A third set of amendments would prevent sanctions relief until they meet goals the U.S. established at the beginning of the negotiations. Critics of the talks claim the administration has backtracked and agreed to too many concessions for Iran.

Republican Sen. Bob Corker of Tennessee, the committee chairman and a co-author of the bill, said he too would like to see Iran change its behavior and he wants any final deal to be a good one that will prevent Iran from developing nuclear weapons. But he said that's not what the bill is about.

"This bill is about the process," Corker said. "It's not a bill about the content of any deal, and hopefully, that's how the bill will remain."

The Associated Press contributed this report.