Sen. Mitch McConnell (Photo by Gage Skidmore via Flickr)

"I've decided I cannot support the treaty," McConnell told CNN's Candy Crowley on "State of the Union." "The verification provisions are inadequate, and I do worry about the missile defense implications of it."

The Senate is expected to continue to discuss the treaty Sunday after an amendment to the treaty, which would have altered language in the treaty's preamble and required Russia and the U.S. to renegotiate, failed to get enough votes on Saturday.

Arizona Sen. Jon Kyl also said that he opposes the treaty and will not vote for its ratification in its current form.

“This treaty needs to be fixed," he told Chris Wallace on Fox News Sunday. "We’re not going to have time to do that.”

Sen. Lindsey Graham (R-S.C.) echoed Kyl's sentiments on CBS's "Face the Nation" Sunday, saying the Senate should wait until next year to vote on START and that he wants to hear from Russia before he will vote for its ratification.

"I’m not going to vote for START until I hear from the Russians that they understand we can develop four stages of missile defense and if we do, they won’t withdrawal from the treaty,” said Graham Sunday on CBS's Face the Nation.

Despite the opposition, some congressional leaders say the treaty will pass the Senate.

"I believe it will pass, and I believe there will be a vote," said Sen. John Kerry (D-Mass.) on ABC's "This Week."

Joining Kerry in his optimism, though with some trepidation, is Sen. Dick Lugar (R-Ind.), who also appeared on ABC's "This Week" Sunday.

Lugar argues that there are enough Republican votes to pass START in the Senate, but he thinks time for the lame-duck Congress is quickly running out.

"The problem is really getting to that final vote," he said. "I think we still have a good number of amendments to be heard and we will be doing out work shortly today."

Reach Deputy Editor Jessika Walsten here.