The Scotland midfielder discussed his experience of Italian football, his relationship with teammate Scott McTominay and the changes brought by Allegri during an interview with Premier Sports TV.
Gilmour said the opportunity to join Napoli was difficult to turn down when the club made its interest clear.
“Serie A is an exciting league,” he said. “When big Italian clubs contact you saying they want to sign you, it is very difficult to say no.”
The midfielder had wanted the move and saw Napoli as an opportunity to compete for major trophies.
“I wanted to come here, I wanted to fight for trophies and be at such a prestigious club,” he said.
Gilmour is also conscious of the history of Scottish players in Italian football. Denis Law is among those who have previously made the journey, while the midfielder believes more Scottish players are now interested in experiencing Serie A.
The transition has also meant adapting to a new tactical environment.
Napoli changed coach during the summer, with Allegri replacing Antonio Conte, while Gilmour has also had to adjust to a new manager with Scotland.
The midfielder accepts that changing systems is part of the process.
“There is a new manager both at the club and with the national team, so you have to get used to new systems and new ways of playing,” he said. “But overall it is exciting.”
One constant in Gilmour's Napoli experience has been his relationship with McTominay.
The two Scotland internationals have become important figures for club and country, and Gilmour has noticed the way McTominay's confidence has grown since moving to Italy.
“Since he has been here, McTominay's confidence has grown enormously,” Gilmour said.
The midfielder believes that confidence is visible in the way McTominay has established himself at Napoli.
“You can see how important he is to the fans and to us as players,” Gilmour said. “We can count on him whenever we need a goal and help. The same goes for Scotland.”
McTominay's importance has therefore extended beyond his performances on the pitch. His presence has helped establish a strong Scottish connection within Napoli, with Gilmour able to see first-hand how the former Manchester United midfielder has settled into Italian football.
For now, however, Napoli are having to deal with his absence.
Gilmour said the squad misses McTominay and is looking forward to having him back.
“We miss Scott and hope everything has gone well,” he said. “We can't wait to have him back with us.”
The change from Conte to Allegri means Gilmour must now continue adapting.
Conte's Napoli had their own tactical identity, while Allegri has brought different ideas and demands. The midfielder accepts that learning another system will take time but sees the adjustment as part of the attraction of playing for a major club.
His comments also underline how different his experience in Italy has become from simply making a transfer to another league.
Gilmour wanted the challenge of Serie A, wanted to compete for trophies and wanted to experience life at a club with Napoli's history. Now he is learning under another of the most experienced coaches in Italian football while continuing to develop alongside McTominay.
The midfielder has also become part of a wider Scottish presence in Serie A, following a tradition that stretches back to players such as Law.
For Gilmour, that history adds another dimension to a move that he already regarded as difficult to resist.
The next stage is adapting to Allegri's Napoli and finding his place in a new tactical structure.
As Gilmour put it, there are new systems to learn and new ways of playing to understand.
For the Scotland international, that challenge is part of what makes the experience exciting.