Drama over immigration could return to the Senate floor next week, as two Republican senators and presidential hopefuls ratchet up their battle over an issue that has sharply divided the GOP field.

Senate Majority Leader Mitch McConnell (R-Ky.) has told top Democrats that he may tee up a vote next week on “Kate’s Law,” according to Democratic aides. Republicans said they gave Democrats a list of bills the GOP wants to take up, including the immigration bill, but no specific timelines were given. The bill, championed by Sen. Ted Cruz (R-Texas), tries to crack down on illegal immigration by leveling tough mandatory minimum penalties on immigrants who repeatedly enter the United States illegally.

A McConnell spokesman said no legislation has been scheduled yet, but noted that the majority leader has previously indicated that "Kate's Law" — which would create a five-year mandatory minimum prison sentence for immigrants here illegally who try to re-enter the country after being deported — would come up at some point.

The legislation is named in honor of Kate Steinle, a San Francisco woman who, authorities say, was shot and killed in July by a Mexican immigrant with a lengthy felony rap sheet who repeatedly entered the country illegally. Steinle’s death prompted a national uproar over so-called sanctuary cities, where local officials decline to cooperate with federal immigration authorities, because San Francisco authorities decided not to notify immigration officials when the suspect, Juan Francisco Lopez Sanchez, was being released from a local jail.

Cruz and other backers of the legislation say the measure is necessary to punish repeat offenders of immigration law. But Senate Democrats are sure to filibuster any attempt from Republicans to advance Kate’s Law, as they did last month on a broader package that targeted sanctuary cities.

“It is sad that the Democratic leader chooses to stand with violent criminal illegal aliens instead of the American citizens,” Cruz said last week as Senate Minority Leader Harry Reid (D-Nev.) blocked the Texas Republicans’ attempt to bring up and pass Kate’s Law.

Aside from Cruz, Fox News host Bill O’Reilly has been particularly vocal in pushing the Senate for a vote on Kate’s Law.

"Everyone knows that a beautiful 32-year-old woman walking in a public place gets killed by a guy who had seven felony convictions here, deported five times and came back six. And you don't want to incarcerate this man in a mandatory way?” O’Reilly said last month. “How can you live with yourself by opposing that law? I feel very strongly that this has to happen to send a message to President Obama and everybody else. You have got to stop this.”

A vote on Kate’s Law, which has been a signature part of Cruz’s immigration platform this year, could bolster the Texas senator at a time when he has cranked up his sparring with Sen. Marco Rubio (R-Fla.) in the GOP presidential primary.

The potential vote was first reported by The Associated Press.