RANCHO MIRAGE, Calif. — President Barack Obama declared Tuesday that Republicans have no constitutional grounds to refuse to vote on a Supreme Court nominee, and he challenged his political foes in the Senate to rise above the “venom and rancor” that has paralyzed judicial nominations.

As Obama cast the dispute over filling the seat of the late Justice Antonin Scalia as a test of whether the Senate could function, there were early signs that Republican resistance could be eroding. Senate Judiciary Chairman Charles Grassley suggested he might be open to considering Obama’s yet-to-be named nominee, an indication his party may be sensitive to Democrats’ escalating charges of unchecked obstructionism.

“I intend to do my job between now and January 20 of 2017,” Obama told reporters at a news conference. He said of the nation’s senators, “I expect them to do their job as well.”

Obama was in California for a meeting of Southeast Asian leaders gathered for two days of diplomacy. But his attention was divided at that conference.

Since Scalia’s unexpected death at a Texas ranch on Saturday, White House lawyers and advisers have been scrambling to refine and vet a list of potential replacements, while also devising a strategy to push a candidate through the Republican-led Senate.

Senate Majority Leader Mitch McConnell has said he doesn’t think Obama should be putting a candidate forward. The Kentucky senator, as well as several Republicans up for re-election this year, say Obama should leave the choice up to the next president. The November election, they argue, will give voters a chance to weigh in on the direction of the court.