The Switzerland international joined the Reds in 2018 but has remained on the fringes of the first team throughout his time at Anfield.
And after enjoying Euro 2020 success, in which he became Switzerland's outright top scorer at major tournaments, Shaqiri has suggested his time under Jurgen Klopp may be coming to an end.
"Fortunately, football has helped me achieve many of my dreams," Shaqiri said. "I was lucky enough to play with great teams and to win many titles.
"Who knows, maybe I could get something even with the Swiss national team. My contract [with] Liverpool expires in 2022, in 2018 I signed a four-year deal.
"The important thing at this moment in my career is to be able to play regularly, but that hasn’t always been the case in the last three seasons.
"This is why I told Liverpool that I feel ready for a new challenge. They accepted my wish and will seriously consider the offers that will come. They will not stop me."
Opportunities have been limited for Shaqiri at Liverpool as he has made just 45 top-flight appearances across a three-year spell, during which he has scored seven goals for the 2019-20 Premier League champions.
But the 29-year-old, whose 10 chances created at Euro 2020 were three more than Ricardo Rodriguez for the Swiss, proved his quality as he was one of Vladimir Petkovic's standout performers at the tournament.
While Shaqiri explained there had been no direct contact from clubs yet, he made clear his desire to keep playing regularly and praised the development of Italian football.
"I'm only 29 years old, I have played in some of the best leagues in Europe, and I would like to continue being a part of them," Shaqiri said. "I’m not yet thinking about the end of my career, it’s a future I imagine far away. We will see what destiny has in store for me.
"I think it’s not just my opinion that football in Italy has developed substantially in recent years. The win at the European Championship is there to prove it, a well-deserved result.
"Just take a look at how many Premier League players have moved to Serie A."
The former Bayern Munich winger, who has been directly involved in 51 goals in 96 appearances for his country, has limited experience in Serie A, too, given he spent six months with reigning champions Inter in 2015.
Asked whether a move to Italy and, more specifically, Lazio was an option, Shaqiri spoke glowingly of Sarri's qualities and complimented sporting director Igli Tare.
"Igli Tare has been doing an excellent job at Lazio for years," Shaqiri commented. "If it's true that he has great respect for me, then I’m honoured.
"I have been following Lazio for a long time, they are a top team. In general, due to my characteristics, I like to play more attacking football and Sarri practices it. It would be intriguing."