"Chris McDaniel won a sizable majority of the votes from Republicans who voted in the run-off," Cruz said in a Monday interview with conservative radio host Mark Levin. "The DC machine spent hundreds of thousands of dollars urging some 30,000-40,000 partisan Democrats to vote in the run-off which changed the outcome."
When pressed by a reporter at the Capitol on Tuesday for evidence of his claim that Sen. Thad Cochran won because of Democratic votes, Cruz said: "The evidence are the numbers from the race."
"The votes that decided that primary were from Democrats, who are all but certain to vote Democrat in the general election," said Cruz, a Texas Republican, weighing in on the Mississippi fight for the first time.
Cochran, who has been in Washington for more than four decades, defeated McDaniel in the June 24 runoff by about 6,700 votes. The McDaniel campaign has alleged Cochran's victory is the result of Democrats who voted in both the Democratic primary and the Republican runoff.
Under Mississippi's open primary system, residents can participate in either primary election, but they are not allowed to vote in both.
While Senate Minority Leader Mitch McConnell isn't disputing the outcome of the run-off contest, he said Tuesday "it would be up to the people of Mississippi to determine the outcome."
"I think it's pretty clear who won," McConnell said Tuesday. "Senator Cochran ran a very successful runoff campaign and got the most votes. Anyone is entitled to contest the outcome and that may well happen in Mississippi."
On Monday, McDaniel's lawyer called for a new runoff election for the GOP nomination, days after the campaign offered a cash prize for any evidence of voter fraud.
ABC's Jeff Zeleny contributed to this report.