The Reds have made a move for the Swiss star, who is free to leave relegated Stoke if his £13m release clause is activated.
Shaqiri is believed to be keen on a move to Anfield but could be offered the chance to return to Italy.
The 26-year-old struggled at Inter Milan earlier in his career but Calcio Mercato reports Lazio are ready to offer him a move.
His wage demands mean the Italian side may well struggle to match Liverpool.
Jurgen Klopp is looking to make the former Bayern Munich winger his new signing of the summer.