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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The start of the 2017-18 Premier League season is just around the corner and is shaping up to be one of the most competitive yet.
As always it is the so called “big clubs” that have splashed the most cash in the transfer window so far this summer. Manchester United or City? Arsenal or Tottenham? Who are the winners and losers of the transfer window so far?