The incident happened after 27-year-old Cipriani was informed that he had not made the final 31-man squad for the tournament but before it was announced publicly five weeks ago.
Catt, 44, threatened Cipriani twice and told him he would "end his England career" before other players stepped in to separate the pair, the Daily Mail reported.
The Rugby Football Union played down the seriousness of the altercation.
"Danny and Mike were involved in a robust conversation on the training pitch following a misunderstanding around a training drill instruction," an RFU spokesman said on Wednesday.
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"Both shook hands afterwards and the incident played no bearing on selection."
Cipriani, a gifted and exciting attacking player but whose defensive qualities have been questioned, has represented England 14 times since making his debut in 2008.
His omission from the World Cup squad after playing well in England's warm-up matches provoked considerable debate and coach Stuart Lancaster was criticised for favouring more defensive players, including rugby league convert Sam Burgess.
England, knocked out of the tournament after losing to Wales and Australia, play Uruguay in their final Pool A game on Saturday.
(Reporting by Ed Osmond; Editing by David Goodman)
