Veach said on Kansas City radio station 810 WHB on Thursday that he was worried enough about the franchise quarterback risking an injury that he put in a call to Mahomes' agent, Leigh Steinberg.
"The [Chiefs] Kingdom can be assured: No more basketball for Pat. ... We were able to nip that in the bud,'' Veach said.
Mahomes was a star in basketball as well as baseball and football in high school, in Whitehouse, Texas. Veach said he was concerned that Mahomes' competitive nature wouldn't allow him to play basketball at something less than full effort.
"With Pat, everything he does is so competitive. ... He actually was in the office yesterday and we were joking about it. He was just shooting around, and all of a sudden it turns into a spin move and juking guys out. He doesn't have that filter in his mind to just have fun and not take everything so competitively," Veach said.
Later at a news conference, Veach called his conversation with Mahomes on Wednesday "a lighthearted discussion."
"I just said, 'Hey, man, you're breaking a lot of ankles out there. Just make sure you don't break your own. I don't think we need that,'" he said.
Mahomes was the NFL's 2018 Most Valuable Player and Offensive Player of the Year in his first season as a starter. He was the second player in NFL history to throw 50 touchdown passes and for more than 5,000 yards in the same season.