An earlier report from the Chilean newspaper La Tercera claimed the 28-year-old has urged Pizzi to give him a day off on 30 August in order to complete "personal formalities" related to his club future.
Sanchez has less than a year left on his contract at the Emirates and has already said no to a new deal. City manager Pep Guardiola is keen on bringing the Chilean international to the Etihad before the transfer deadline and Arsenal have already rejected a £50m ($64.6m) bid from City for the attacker.
Chile take Paraguay in the World Cup qualifiers on 31 August and the Gunners forward is currently on international duty with the national team. Pizzi has denied speculation suggesting Sanchez has requested to leave the training camp to sort out his future.
"No. It's absolutely false. That permission never existed. Sometimes players request authorisation for some things and sometimes we grant them and other times we don't," Pizzi explained, as quoted by emol.
"I'm against rigid decisions, each case is different. We consider the individual situation of each player regarding many variables. But we always take into account that we are in a national team camp. He is very, very good. He is happy.
"Until the deadline for transfers is closed, it's difficult to elucidate Alexis' future. In any of the possible options that are being handled, his future is very good. Then there is no reason for him to be dissatisfied or annoyed."
"Any of the options he has are very good or excellent. We are talking about a top player that plays for one of the best teams in the world and is trained by one of the best coaches in the world.
"To say that he can be nervous about his future is only a metaphoric way to talk about this. All of us would want to have this kind of worry. In many occasions, the decisions don´t depend on one person, there are other interests involved and the player cannot control them. He is at the expense of this decisions."
Sanchez scored 30 goals and registered 15 assists in the 2016/17 season. He played a key role in the north London club win their third FA Cup in four years, but failed to help Arsene Wenger's side qualify for the Champions League.