U.S. Sen. Ted Cruz (R-Texas) invited Donald Trump to the U.S.-Mexico border. (Photo by Mark Wilson/Getty Images)

Sen. Ted Cruz (R-Tex.) invited Donald Trump to tour the U.S.-Mexico border last month, though the senator couldn't make it in the end because of Hill votes, a Cruz spokesman said Wednesday.

"We were talking about immigration and then Ted said to Mr. Trump, ‘you should come to the Texas border. Be my guest and we’ll go together,’” Cruz spokesman Rick Tyler said. The invitation was extended during a meeting between the two men at Trump Tower in New York. He said the campaigns had a date set, but Cruz couldn’t go because the Senate had votes scheduled.

Trump, who did make a whirlwind visit to the U.S.-Mexico border last month, released a position paper on immigration this week that calls for, among other things, deporting the undocumented en masse and denying citizenship to their U.S.-born children. He has repeatedly called for a wall to be built between the two nations. His campaign did not immediately respond to a request for comment about the Cruz Team's claim.

"I’m glad he’s in the race and he manages to attract a lot of attention and actually to force the media to talk about issue that the media should have been talking about, in particular illegal immigration," Cruz said of Trump Wednesday on the Michael Medved Show. "I welcome Donald Trump's immigration proposal. Virtually every element in the proposal he submitted is contained within legislation that I filed years ago in the Untied States Senate,” Cruz said, adding he is glad to see Trump is calling for tripling the number of border patrol agents.

[Read: Donald Trump makes a Texas-size splash with visit to Mexican border]

Cruz and Trump, though political rivals, have been chummy and occasionally downright laudatory toward one another in the past few months. On Wednesday in New Hampshire, when asked about a running mate, Trump mentioned Cruz.

"I like Ted Cruz. I like Dr. [Ben] Carson. They’ve been very supportive of me," Trump told NH1.

At times, the friendship has felt a bit lopsided. When Cruz met with Trump at Trump Tower last month, the mogul sounded a slightly less-than-effusive note.

“Ted Cruz called me,” Trump said on MSNBC’s Morning Joe. “And I don’t know why I’m meeting him, to be honest, but I do have respect for him.”

It is on the issue of immigration where Cruz and Trump have found the greatest common ground.

Trump's controversial remarks about Mexican immigrants early in his campaign led Macy's and Univision to sever ties with him. At the time, Cruz defended Trump, calling the businessman "bold" and "brash."

"I salute Donald Trump for focusing on the need to address illegal immigration," Cruz said on NBC's "Meet the Press."

Tyler said the men have a “very cordial, friendly relationship” and genuinely like one another. “They really enjoy each other’s company,” Tyler said.

It isn't just about the friendship -- Cruz is hoping to get something a bit more tangible from Trump as well: his supporters.

"We want to win over all the supporters of other candidates," said Tyler -- and “[Trump] has the most voters that we hope to eventually win their support."