Mr. Walker’s shifts on issues this year have created friction with a variety of people open to supporting him. He used to oppose what he called government mandates on the use of ethanol in gasoline, for example, but told Iowans this year that he was willing to continue one, the Renewable Fuel Standard. The reversal was not well received in the political network led by the industrialists David H. and Charles G. Koch, according to a Republican aware of the reaction who spoke on the condition of anonymity because of sensitivities over the group’s deliberations.

But his stance on marriage is what has disquieted people who had counted on Mr. Walker taking a more restrained approach to the culture wars.

For several months, according to four people briefed on the discussions who were not authorized to describe an off-the-record meeting, Republican donors who supported legalizing same-sex marriage had worked quietly to try to build bridges to Mr. Walker, whose wife has a lesbian cousin whose wedding reception Mr. Walker attended.

The donors were cheered by a remark Mr. Walker made in the fall when he was locked in a hard-fought re-election battle. Asked about same-sex marriage after his state’s ban was struck down in federal court and the Supreme Court refused to review that decision, Mr. Walker said, “For us, it’s over in Wisconsin.”

The remark was also the subject of much critical discussion among social conservatives, according to one leader of that faction of the party, who was granted anonymity to describe private conversations.

At a gathering of Republican donors in New York in the spring, Mr. Walker indicated that his response to an eventual Supreme Court ruling, if it deemed same-sex marriage constitutional, would be in keeping with the spirit of his earlier remark about the question being a settled one in Wisconsin, people who attended the meeting said.