Republican Donald Trump has called for a “one-on-one” battle with Ted Cruz, urging other rivals to quit the nomination race.
Ted Cruz also suggested it was time for other hopefuls, like Marco Rubio and John Kasich, to step aside.
Donald Trump and Ted Cruz won two states each in March 5 voting.
In the Democratic race, Bernie Sanders took two states – but Hillary Clinton maintained her front-runner status after a big victory in Louisiana.
Speaking after wins in the Republican Kentucky caucuses and Louisiana primary election, Donald Trump told a news conference: “I would love to take on Ted Cruz one on one.”
“Marco Rubio had a very very bad night and personally I call for him to drop out of the race. I think it’s time now that he dropped out of the race. I really think so.”
Meanwhile, Texas Senator Ted Cruz – who won Republican caucuses in Kansas and Maine – said he believed that “as long as the field remains divided, it gives Donald an advantage”.
Florida Senator Marco Rubio and Ohio Governor John Kasich have so far made no public comments on the issue.
Ted Cruz now appears to be the only candidate who can stop Donald Trump, analysts say, after a week in which the Republican establishment did everything it could to attack the New York billionaire.
Observers also point out that Ted Cruz and Vermont Senator Bernie Sanders – who beat Hillary Clinton in Kansas and Nebraska – both won in states holding caucuses but lost in the Louisiana primary elections, involving far more voters.
They say that the Louisiana race also appears to have exposed Bernie Sanders’ lack of support among African-American voters.
Hillary Clinton said she was thrilled to add to her delegate count.
The former secretary of state said: “No matter who wins this Democratic nomination, I have not the slightest doubt that on our worst day we will be infinitely better than the Republicans on their best day.”