It happened in the 17th over of the MI innings when RCB's Starc continued with his delivery despite the batsman pulling out of his strike.
As the Aussie fired the ball at Pollard, the batsman advanced menacingly towards Starc and flung his willow at the bowler but, luckily, it slipped out of the Trinidadian's hand and dropped near him.
A fuming Pollard had a word with the umpire complaining about the bowler's conduct.
After the bat lay on the turf, the two on-field umpires intervened to talk to the two angry players as Chris Gayle, playing for RCB, came on to calm his West Indian teammate.
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The on-field umpires S Ravi and K Srinath also had a word with RCB captain Virat Kohli.
Later, in the final over, following a misunderstanding, Pollard and Rohit Sharma were at the same end and bowler Starc clipped off the bails. Pollard kicked the ground in disgust before leaving.
The matter is expected to be taken up by match referee Andy Pycroft after the game and action could be taken against both Pollard and Starc.