The England international goalkeeper is wanted by Arsenal as well as the Merseyside outfit.
But the Daily Mirror report that Butland would favour a switch to Anfield, rather than head to Arsenal due to Jurgen Klopp's influence.
A lack of quality between the posts has cost Liverpool dearly in recent months.
And boss Jurgen Klopp is determined to fix the issue once and for all when the summer transfer window rolls around.
Loris Karius has established himself as German's No.1 in recent weeks but he is still prone to costly errors and an upgrade is needed.
Roma's stopper Alisson has been touted but the Brazilian would cost significantly more.
Butland, 25, has impressed at Stoke this season and could well be England’s No.1 at this summer’s World Cup finals.
Simon Mignolet's Liverpool career looks to be over and the Belgian will be moved on at the end of the season.
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Klopp deems Premier League experience vital in goal. Butland has played 68 top-flight games for the Potters.
Karius will remain at Anfield as second choice according to The Mirror's report.
Butland faces Manchester City's awesome attack at the Bet365 Stadium tomorrow night tomorrow night (8pm).