Senate Majority Leader Mitch McConnell (R-Ky.). (Timothy D. Easley/AP)

Senate leaders are rewriting their health care plan in an effort to vote on it next week, Republican Whip John Cornyn (R-Tex.) said Monday, even as some GOP senators expressing deep pessimism about the prospect of reaching a final agreement.

The push for a revised bill comes as Senate Democrats are working to enlist the help of Republican governors to scuttle the current health-care proposal. Some rank-and-file Republicans have suggested their party should negotiate with the minority.

Cornyn said that he expects GOP leaders to unveil a new version of the legislation this week, “and then we’ll vote on it next week.”

At their normal weekly policy lunch Tuesday, Republican senators are expected to hear how their concerns have been addressed, and leaders can measure whether their tweaks are likely to move the needle at all, according to a GOP aide familiar with the talks.

A spokesman for Senate Majority Leader Mitch McConnell (R-Ky.) declined to say when he will release the new version of the bill.

While the prospect of a compromise between the two parties on overhauling the Affordable Care Act may prove daunting given the ideological divide between Republicans and Democrats on health care, the ongoing conversations among a handful of senators suggests some lawmakers are…