LAS CRUCES, N.M.  Senator John McCain is facing increasing scrutiny about his selection of a running mate as some social conservatives expressed alarm on Tuesday that Mr. McCain might ask a candidate who favors abortion rights to join him on the ticket.

But other conservatives said that Mr. McCain, who has long been in step with the Republican Party platform in opposing abortion, was unlikely to be the first Republican presidential nominee in decades to select such a candidate. They said that Mr. McCain’s recent public flirtation with Tom Ridge, a former Pennsylvania governor who supports abortion rights, was as much to give the appearance that Mr. McCain had an open mind on the issue as it was an embrace of Mr. Ridge.

“I think there’s such sufficient understanding at the most senior levels of this campaign of the importance of the pro-family constituency that I’m very comfortable with how the selection process is proceeding,” one Republican strategist close to the campaign said. “I think social conservatives will be fine.”

Republicans said Mr. McCain would name his choice after the Democratic convention ended next Thursday. Mr. McCain’s aides said that no day had been set, but that one strong possibility was Friday, when Mr. McCain has a rally scheduled in Ohio.