The 25-year-old Brazil international said he was "very happy" to stay at the club as he prepared for the World Cup, but rumours about his future persist with talk of a move to Real Madrid emerging as a reported possibility.
When asked whether Alisson might be joining Napoli, Pallotta was quoted as telling AreaNapoli: "No, [Napoli president Aurelio] De Laurentiis doesn't want to spend all his money. €78 million is needed and he's close to joining Real Madrid."
The Roma chief has moved to clarify his comments, which he said were made in jest.
"I have read with much amusement the quotes attributed to me about the transfer market," he said in a statement on the club's official Twitter account.
"In fact I was joking around with a group of guys and responded sarcastically with a reference to the popular rumours on Twitter and in the papers since I've been in Italy.
"There was nothing serious about anything I said whilst making pizza. Just stupid American humour that some don't get."
President Pallotta: "I have read with much amusement the quotes attributed to me about the transfer market. In fact I was joking around with a group of guys and responded sarcastically with a reference to the popular rumours on Twitter and in the papers since I’ve been in Italy." pic.twitter.com/3p4U7ie9L1— AS Roma English (@ASRomaEN) June 14, 2018
Alisson has previously declared he would like his future determined before the start of the World Cup.
However, if he is to move, then that wish appears unlikely to occur, as Brazil open their World Cup play in Group E against Switzerland on Sunday.