The Slovakia international has been regularly linked with a move to Old Trafford over recent seasons - but there's a belief that Napoli could be tempted to sell their highest-profile midfielder, who has fallen down the pecking order under current manager Rafael Benitez.
Hamsik has completed just three Serie A matches in 2014, and failed to even get off the bench during Sunday's 4-2 home victory over Lazio.
It was reported in March that Napoli president Aurelio De Laurentiis still regards Hamsik as an integral and indispensable member of the Azzurri squad, but Benitez's use of the 26-year-old appears to have encouraged Manchester United to step up their pursuit of the player.
A €31m offer could tempt De Laurentiis into reluctantly agreeing to sell Hamsik, who has scored 76 goals in 297 appearances for Napoli since making his debut in 2007. But an offer of €15m plus Kagawa, who has rediscovered some form and confidence in recent weeks, may prove even more tempting for the Serie A club.
Hamsik for Kagawa would be a like-for-like replacement for the reigning Premier League champions, who hope a squad overhaul this summer will leave the club in a better position to compete for silverware next season.
Manchester United fans: should the Red Devils offer cash and Kagawa for Hamsik? Have your say in the comments section below.