Story highlights

NEW: Sen. Lisa Murkowski tweeted that President Barack Obama should allow his successor to pick the next Supreme Court justice

The Alaska Republican had said she thinks Obama's pick to replace Antonin Scalia should get a hearing

Most other Republican senators have said a nomination should wait until after the presidential election

Washington CNN  — 

Alaska Republican Sen. Lisa Murkowski said the next president should choose the Supreme Court Justice to replace Antonin Scalia, days after stating whomever President Barack Obama picks should get a hearing in the Senate.

“I urge Pres. Obama to follow a tradition embraced by both parties and allow his successor to select the next Supreme Court justice,” the Alaska Republican tweeted Thursday evening.

“If POTUS ignores precedent, I believe extraordinary circumstances give the Senate every right to deny the nominee an up or down vote. (5/5),” she added on Twitter.

Originally breaking with many of her Republican colleagues, Murkowski was the first GOP senator to say a Supreme Court nominee from Obama “should get a hearing.”

“I do believe that the nominee should get a hearing,” she told reporters In Juneau on Wednesday, according to the Alaska News Dispatch. “That doesn’t necessarily mean that that ends up in a vote. The purpose of the hearing is to determine whether or not this individual, based on their record should be named to the highest court in the land.”

Most other Republican senators who have spoken publicly about replacing Scalia, a leading conservative on the high court, have said a nomination should wait until after the election and a new president is in office.

“The American people‎ should have a voice in the selection of their next Supreme Court justice. Therefore, this vacancy should not be filled until we have a new President,” Senate Majority Leader Mitch McConnell said Saturday shortly after Scalia’s death.

Senate Judiciary Committee Chuck Grassley, R-Iowa, who initially said a nominee should wait, has suggested since he might be open to allowing confirmation hearings.

Democrats blasted Republicans for their defiant stances. But Senate Republicans are growing increasingly comfortable with the idea of not allowing a hearing at all and standing by their reasoning that the Democratic president should not be able to greatly alter the ideological makeup of the court so close to the end of his term and during one of the most tumultuous presidential campaigns in recent history.

But refusing to even consider a nominee runs the risk that not all GOP members will be on board with the strategy, as the comments from Murkowski, who often works across the aisle and who is running for re-election this year, seem to reflect.

Sen. Thom Tillis, R-North Carolina, said this week Republicans could appear as “obstructionists” if they reject a nominee outright. Other GOP senators have hinted at the possibility of holding hearings once Obama announces a nominee but not nearly as strongly as Murkowski.

Two moderates Republicans, Susan Collins of Maine and Mark Kirk of Illinois, have said it’s too soon after the death of the long-serving justice for a political debate over his replacement.

But Collins added a little more to that when she told an audience in Maine Wednesday that once a nominee put forward senators “should carry out our constitutional duty,” according the Portland Press Herald.

The Senate returns from a week-long recess Monday and Republicans are expected to hold their first in-person strategy session over the issue on Tuesday.

A White House spokesman suggested this week the President could name a nominee as early as next week.