Sen. Lamar AlexanderAndrew (Lamar) Lamar AlexanderGOP lawmakers want Trump to stop bashing Congress The Hill's Morning Report - Sponsored by CVS Health - A pivotal day for House Republicans on immigration GOP, Dem lawmakers come together for McCain documentary MORE (R-Tenn.) chided Democrats for canceling a markup on the bill that would fund ObamaCare in 2015.

“Our markup was indefinitely postponed ... because some senators don’t want to vote on difficult or tough amendments,” Alexander said on the Senate floor Thursday. “That’s what we do.”

The Labor, Health and Human Services appropriations bill was supposed to be marked up in the full Senate Appropriations Committee this week, but Sen. Tom Harkin (D-Iowa) postponed it indefinitely.

Harkin denied that leaders have taken the bill off the schedule because vulnerable Democrats feared it would provoke "gotcha" amendments that could hurt their reelection chances in November.

But Alexander said that was precisely the reason the hearing was canceled.

He said he hoped to offer at least four amendments at that hearing, one of which would have required the administration to provide weekly reports on who is enrolling in the ObamaCare health exchange.

“It’s very simple and it shouldn’t be controversial or a ‘tough’ vote for any senator,” Alexander said. “It would simply require the Obama administration to provide weekly reports during open enrollment to Congress, states, and the public about the federally-run exchange for 36 states.”

Alexander’s other amendments sought to limit the powers of the Department of Education and the National Labor Relations Board. He said he hopes the markup will be rescheduled so that he can get a vote on his proposals.

"The Democratic gag rule on important amendments has moved from the Senate floor to committees. This is a new level of obstruction," Alexander said. "I hope to have the opportunity to offer these amendments at a markup."

— Erik Wasson contributed to this article.