Relegation-threatened Livorno have sacked Domenico Di Carlo after just 88 days in charge and re-appointed Davide Nicola.
Virgil van Dijk claims he knows nothing of interest from English clubs and is keen to prove himself in the Champions League with Celtic.
The England Under-21 midfielder has already made five starts under Swans boss Michael Laudrup since heading to South Wales in a £5million transfer in early July.
Shelvey was only included in Brendan Rodgers' starting line-up for Liverpool games in the Premier League nine times last season. He made another nine starts in Europa League and domestic cup fixtures, but with Reds captain Steven Gerrard his main rival for an attacking midfield berth, he appreciates the competition was fierce.
"It's a big thing, trying to knock Steven out of the team," Shelvey told the Daily Mail.
"If you had a really good game, the next week you might still be out because the likes of Gerrard are coming back. Imagine that. And you can't argue with it - it's crazy how good he is."
Despite that, Shelvey says he was "pretty excited" when he reported back to Melwood for pre-season training - but the same evening, his agent called to tell him about Swansea's interest.
Shelvey added: "Brendan told me I could stay there and see what happened. It really was my choice. I wasn't pushed.
"But I wanted to play football. I would hate to be one of those sitting on my money. It is in my character, the way I was brought up. Even Brendan said, 'You're not one of these squad players happy to stand around'."
The 21-year-old says he was given positive signs over how many minutes he would get under Laudrup - "he told me if I play well in a game, then I will start the next game" - and having won his first senior England cap last summer, he believes he can recapture the acclaim that he once enjoyed.
"It's about confidence and knowing you'll get game time if you do well," said Shelvey, who is set to line up his former club when Swansea host Liverpool at the Liberty Stadium in Monday Night Football.
"I wanted to play, to get back the status that I used to have. Now I am playing."