The Daily Star says Saints see Clasie as a replacement for wantaway midfielder Morgan Schneiderlin.
Clasie will command a fee of around £10m.
Feyenoord previously wanted £13m for Clasie but will settle for £10m to speed up a transfer, with Sunderland also keen.
Koeman has previously said he would like Clasie to succeed Schneiderlin in the Saints midfield, after managing the Dutch international during his three-year stint at the Dutch top-flight side.
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