The Sun reports that the two Premier League giants are keen on signing the 26-year-old midfielder.
Krychowiak has become a key player at Sevilla and Arsenal boss Arsene Wenger wanted to sign the midfielder in the January transfer window.
Wenger will make a new move for Krychowiak this summer as both Mikel Arteta and Mathieu Flamini looks set to leave the Gunners at the end of the season.
Arsenal face competition from the likes of Manchester City, Real Madrid and Juventus.
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Liverpool have unveiled their new home kit for next season, sticking with the hugely popular retro styling of the current shirt.
The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.