The Rams boss is unhappy that the Ireland boss has been urging his players to seek moves away from the club and said that it is not the Italian's place to comment
Praising the quality of Rams players Jeff Hendrick and Conor Sammon, the Republic of Ireland manager suggested that the duo should seek moves to the English Premier League.
However, Derby boss Clough criticised the former Italy and Juventus manager for his comments and said that he should not comment on anything other than the players' performances.
"We don't really think it is his place to be commenting on things like that," he told the BBC. "Just talking about the form of the players would be enough."
It is not the first time that Trapattoni has incurred the ire of a club manager, with newly-instated Wigan Athletic boss Owen Coyle recently complaining that the Italian tactician was unsettling James McCarthy by stressing that the player needed to seek a transfer.
Nevertheless, despite Nigel Clough's annoyance, Derby County chief executive Sam Rush attempted to put a positive spin on Trapattoni's comments.
"I will take the comments in the positive sense, that Mr Trapattoni was talking about how good the Derby County players were," said Rush. "I'm not sure if some things were lost in translation — some of it was quite literal, so perhaps it wasn't.
"We're pleased that people think highly of Derby County players but we're in no rush to put them in the Premier League with anyone other than Derby County."