Rep. Jim Jordan, Republican of Ohio, still faces an uphill climb to the House speakership, with at least 10 to 20 Republican members who oppose his nomination, CBS News has learned, based on background conversations over the weekend with six key House Republicans and more than a dozen sources familiar with the deliberations.

“At least 10 to 20,” one of the House Republicans told CBS News on Sunday, while another added that that Jordan’s support has grown incrementally in recent days but remains soft.

While Jordan’s confidants remain optimistic that he can get to the necessary 217 votes Tuesday, when the House is scheduled to bring a vote to the floor, several who are more critical of Jordan privately insisted this weekend that at about a dozen Republicans remain unwilling to support him, due to their frustrations over how Rep. Steve Scalise, Republican of Louisiana, was treated during his speaker bid and their simmering anger over the ouster of former Speaker Kevin McCarthy. They also are wary of whether Jordan can handle the intensity of the challenges facing Congress in the coming months.