Under the Constitution, presidents must be "natural born citizens." Cruz was born in Calgary, Alberta, to an American mother, which he says makes him eligible to run.
Senate Majority Leader Mitch McConnell told ABC's "This Week" the Senate would not act to counter Trump's claim that Cruz's Canadian birth makes him ineligible to be president. The father of the senator from Texas was born in Cuba.
In 2008, the Senate passed a resolution declaring Senator John McCain, a Republican presidential candidate, a natural born citizen. McCain was born to American parents on a US military base in the Panama Canal Zone.
"I just don't think the Senate ought to get into the middle of this," McConnell said. "These guys are all slugging it out in Iowa and New Hampshire. We'll have a nominee, hopefully, by sometime in the spring."
The winner will face the Democrats' nominee in the November general election.
Trump, who leads Republican candidates in national opinion polls, is grappling with the rise of Cruz in Iowa, which holds the first presidential nominating contest next month.
As Cruz took the lead in Iowa before its February 1 caucuses, Trump's glare followed. The billionaire businessman highlighted the citizenship issue last week, warning that Democrats could challenge Cruz's eligibility in court.
Cruz has refused to engage with Trump on the issue. On Sunday, he said the law was straightforward.
"As a legal matter, the Constitution and federal law are clear that the child of U.S. citizens born abroad is a natural-born citizen," he told CNN's "State of the Union."
Asked on "Fox News Sunday" whether he really doubted Cruz was a natural born citizen, Trump said, "I don't know. I really don't know. It depends.
"Does natural born mean born to the land, meaning born on the land? In that case, he's not."
Trump said the term has not been adjudicated, and advised Cruz to seek a judgment.
"He has to solve this problem because the Democrats will sue him if he's the nominee," Trump said on NBC's "Meet the Press."
Another Republican candidate, Senator Rand Paul of Kentucky, said on CBS' "Face the Nation" it was unclear if Cruz qualified as a natural born citizen and that Democrats will likely challenge that.
Cruz said the attacks against him are telling.
"Three weeks ago, almost every Republican candidate was attacking Donald Trump," Cruz said on CNN. "Today almost every Republican candidate is attacking me. And that kinda suggests something has changed in the race."