Both of them, however, offered each other the vice- presidential ticket if they were to bag party's presidential nominee this summer.
At the Republican presidential debate yesterday, the first of the year, Cruz alleged that Trump is now raising the issues because his poll numbers are going up.
Trump agreed.
Cruz said: "I recognise that Donald is dismayed that his poll numbers are falling in Iowa. But the facts and the law here are really quite clear. Under longstanding US law, the child of a US citizen born abroad is a natural-born citizen.
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He added: "I would suggest we focus on who's best prepared to be commander-in-chief, because that's the most important question facing the country."
Trump argued that the questions about eligibility of Cruz because he was born in Canada are being raised by eminent constitution lawyers and it is time that the Texas Senator gets it clarified by the Court or else the Democrats are getting ready to file a lawsuit against him.
"If for some reason, he beats the rest of the field, I already know the Democrats are going to be bringing a suit. You have a big lawsuit over your head while you're running. And if you become the nominee, who the hell knows if you can even serve in office? So you should go out, get a declaratory judgment, let the courts decide," Trump said.
Responding, Cruz said he has spent his entire life defending the Constitution before the US Supreme Court.
"And I'll tell you, I'm not going to be taking legal advice from Donald Trump," he asserted, but later added: "Donald, you very kindly just a moment ago offered me the VP slot. I'll tell you what. If this all works out, I'm happy to consider naming you as VP. So if you happen to be right, you could get the top job at the end of the day."
"Actually, I'd love to get you to build a wall," Cruz responded.