Before leaving for the August recess, Cantor said he wanted to bring up the bill under suspension when Congress returned for its brief eight-day September session.
"Unfortunately we don’t see the bipartisan coalition that we need in order to pass this," he said.
Cantor said the House is continuing to work with the Senate.
Sen. Ben Cardin (D-Md.), author of human rights legislation that is expected to move as part of the trade bill, rejected Cantor's argument.
"We have the votes over here. He has the majority and if he put it out there I'm sure the Democrats [in the House] would support it," Cardin said.
"There's the votes in the Congress to get this done."
On Monday, House Majority Whip Kevin McCarthy (R-Calif.) said the prospect of moving the bill this month was "a little difficult."
"I will tell you the Republicans are ready to go on that," he said.
"The White House has not put the effort in or come up with the number of [votes] they said they would have in the conversations, so I'm waiting on the White House.
"But we're prepared on our side to be able to solve that."
A House aide told The Hill on Tuesday that "we are hoping for a vote next week that avoids amendments and other process hurdles."
Last month, business groups that have been pressing for completion of the legislation said they had expected a bill to come to the floor this week.
U.S. Trade Representative Ron Kirk along with President Obama have said that passing the Russia trade bill — they have been opposed to attaching human rights legislation — is a top legislative priority.
Moscow joined the World Trade Organization on Aug. 23 and proponents argue that lifting the obsolete Jackson-Vanik provision will help U.S. businesses.
Lawmakers plan to add human rights legislation to the trade measure — the so-called "Magnitsky bill," which would punish Russian officials for their involvement in the death of whistleblower Sergei Magnitsky, a lawyer who died in prison after reporting government corruption in Russia.
Congress needs to repeal the 1974 Jackson-Vanik provision, a U.S. law aimed at encouraging the emigration of Russian Jews with the threat of higher tariffs, to establish normal trade ties.
The Senate Finance and House Ways and Means committees each unanimously approved the bill, albeit different versions, in July.
The Senate linked the trade bill to grant Russia PNTR with human rights legislation while the House panel approved the trade bill on its own with the understanding that some form of the Magnitsky bill would be attached.
If Congress can't finish the bill this year now or in the lame-duck session, Cardin said it does hurt and "there's damage done to our businesses.
"But we have the right formula, with Magnitsky and PNTR. By next year things could change. Let's get it done now."
Russell Berman contributed.
By Vicki Needham and Julian Pecquet