Danks made the comments in response to an incident in the fourth inning of Toronto's 13-4 victory when Bautista popped up a 3-2 pitch from Danks for the second out of the inning.
Bautista tossed his bat and appeared frustrated with himself as he jogged toward first base before the ball was caught.
The two players had words with each other as Bautista headed back to the dugout.
"I just told him to run the bases. He was out there acting like a clown," Danks said after the game. "He's had a great year and a half -- no doubt. He's one of the best players in the league. But he's out there acting like he's Babe Ruth or something."
"Just the way he was acting. He ran halfway down the line and stopped and spiked his bat. I get it. He's upset at himself," Danks continued. "He's a good hitter, he's had a great couple years. But he isn't that good to be acting like he needs to hit every ball out of the ballpark."
After the game, Bautista said he was simply frustrated with himself and not trying to show up Danks.
"That should not be his concern," Bautista said of Danks' reaction to the popup. "I was upset at myself with missing a pitch. If he took it the wrong way, I'm sorry. I'm not here to make him feel good."
Bautista did not elaborate on what he said in response to Danks as he jogged back to the dugout and said he wasn't concerned with the incident.
"It really doesn't matter to me what he thought. What I'm not going to allow when I'm running by him is him yelling at me, so I yelled back at him," Bautista said.
Bautista leads the Majors with 20 home runs and has more home runs than any other Major Leaguer since the beginning of 2010 with 74 in his last 205 games.
Danks called Bautista a "good player" but still felt he stepped over a line during the incident.
"I'm not going to let him sit out there and show me up like that," Danks said. "I just told him to run the bases and quit acting like a clown."
Arden Zwelling is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.