"Should I do it just for fun?" Trump asked at a campaign rally here.
"It's so nasty though," Trump added. "Ugh, I'm so good at that stuff."
Trump has repeatedly suggested that Cruz may not be eligible to serve as president because he was born in Canada, raising the question of whether Cruz meets the Constitutional requirement that the president be a natural-born citizen.
Cruz has insisted that he meets that bar because he was born to an American mother, making him an American citizen by birth.
Most legal experts have agreed with Cruz's assessment, but the issue has never been settled in federal court, and Trump noted Saturday that he would have legal standing to sue Cruz over the issue as an opponent for the Republican presidential nomination.
Trump began suggesting earlier this month that Cruz's Canadian birth could endanger Republicans' chances of winning the White House, arguing that Democrats would sue to keep Cruz out of the White House.
Cruz held dual citizenship until he officially renounced his Canadian citizenship in 2014.
Trump on Saturday also reprised a line
he used against Cruz on Twitter just a day earlier, noting that Cruz "could run for the prime minister of Canada."
"And I wouldn't even complain because he was born in Canada," Trump said.
Trump's comments Saturday came nine days before Iowans are set to head to the caucuses to vote in the first contest of the primary cycle.
Trump and Cruz are neck and neck in recent polls, and both candidates have amplified their attacks against each other in recent weeks.
Trump this week launched his first attack ad
against Cruz, suggesting the Texan has been inconsistent on immigration, a contrast with Trump's hardline position on the issue.
Trump on Saturday also raised the subject of Cruz's previously undisclosed loans
he received from Goldman Sachs and Citibank to finance his political campaign that propelled him to the Senate.