It appeared as though a switch to Spain was on the cards at one stage, with James Pallotta suggesting that a €78 million (£69m/$91m) deal was close.
The Roma president has since claimed that he was joking and that those in the Italian capital have no intention of parting with a prized asset.
That remains the stance when it comes to a man currently away on World Cup duty, with Monchi adamant that no rumoured interest has been formalised.
Roma’s sporting director told reporters at the club’s unveiling of new signing Javier Pastore: “As of now we’ve had no offers for Alisson.
“All I think about is waiting for the World Cup to finish, but hopefully for Brazil that’s as late as possible, then he can rest, come here, and train with his team-mates.
“I want to make it clear: as of now no offer has arrived. If you asked me about [Patrick] Schick I’d say the same thing, I’ve read so much but no offer has come in.
“That’s the reality, and I have to work with reality.”
Alisson is not the only Roma player to have sparked plenty of transfer talk, with Lorenzo Pellegrini another to have been linked with leading sides across Europe.
He added: “I met Giampiero Pocetta [Pellegrini's agent] last week, and we did not talk about his goodbye. We are calm.”