Hitoshi Tamura appears to be open to testing the free agent waters. While he would like to stay with the team, it seems he also wants a multi-year deal worth around 100M yen. Since the Hawks usually make it a point to stay away from multi-year deals, the two sides could run into some problems during negotiations.
As for why Tamura is interested in a multi-year deal, according to someone close to Tamura, he loves the city of Fukuoka and loves playing for the Hawks, but hasn't been able to invite his family to live with him because of his contract situation. If Tamura can get a multi-year deal, his family might feel secure enough to make the move to Fukuoka, which would be a big plus for Tamura.
A Softbank exec told reporters that while the team does consider multi-year deals on a case-by-case, their basic rule was to push one year deals.
A number of MLB teams, including the Diamondbacks, Rangers, and Giants, also apparently checked out Tamura during the season.