Mr Cruz also suggested it was time for other hopefuls, like Marco Rubio and John Kasich, to step aside.
Mr Trump and Mr Cruz won two states each in Saturday's 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, Mr Trump told a news conference: "I would love to take on Ted Cruz one on one."
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"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 Mr 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 Mr Rubio and Ohio Governor Mr Kasich have so far made no public comments on the issue.
Mr Cruz now appears to be the only candidate who can stop Mr Trump, analysts say, after a week in which the Republican establishment did everything it could to attack Mr Trump, a New York tycoon.
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