The Switzerland attacker could be available for around £12 million following his relegation with Stoke City last season.
Shaqiri has previously stated his desire to remain in the Premier League, prompting reported interest from a host of top-flight clubs - including Liverpool and Everton.
The Liverpool Echo have reported that Reds boss Jurgen Klopp is interested, while the same publication also claimed that the Toffees are interested.
"At the moment, I can’t say anything,” he told the Mirror [12.24] "You’re going to hear soon on my future.
"I was not thinking about this because I was just focused on my national team and I wanted to make a good tournament for my team and my nation.
"It’s pretty normal to be aware of clubs being interested. Of course, I would like to play on the biggest stage in football, everybody knows that. I hope I can stay in the Premier League. I hope so, but you never know with football. I don’t rule out anything.”
Has he dropped a hint?
Shaqiri's desire to play on "the biggest stage" could suggest that he's referring to the Champions League, in which case Liverpool have a major advantage over Everton.
The Toffees finished eighth in the Premier League last term, while Klopp's side finished fourth but reached the final of Europe's elite competition, only to be beaten by Real Madrid in Ukraine.
If it's Champions League football he's after, he won't be joining the Goodison Park side if their Merseyside rivals are interested.