Saints have made an improved contract offer to Jay Rodriguez, with negotiations expected to continue after the pre-season trip to Belgium.
The club are understood to have tabled enhanced terms to the forward last week, after initially submitting a proposal for a new five-year deal a few months ago.
Discussions began around the time Rodriguez ruptured the anterior cruciate ligament in his right knee, during April’s 4-1 defeat at Manchester City.
Despite the injury, which is expected to rule the 24-year-old out until October, Saints have remained keen to tie Rodriguez down to a fresh, long-term contract.
The England international, who is admired by Tottenham, with Liverpool and Arsenal also reportedly interested, has two years left on his current deal at St Mary’s.
He is set to discuss the improved offer after returning from this week’s trip abroad.
Rodriguez’s current deal dates back to when he signed from Championship club Burnley in 2012, just before Saints began their first campaign back in the top-flight.
His performances and progression since then have merited an improved offer, with the forward scoring 15 Premier League goals last season and earning senior international honours.
Had it not been for his knee injury, Rodriguez may have won a place in England’s World Cup squad this summer.
His rehabilitation is said to have been progressing extremely well and he joined his teammates this week for their pre-season training camp in Belgium.
Rodriguez’s contract is not the only one that Saints must consider this summer.
The club, and Koeman, are also weighing up whether to offer fresh terms to players such as Jack Cork, Jose Fonte, Artur Boruc, Maya Yoshida and Jos Hooiveld, all of who are out of contract next summer.
Saints are also working on adding more pieces to their squad, having already completed the captures of Dusan Tadic and Graziano Pelle.
The club are said to have had an approach for Norwich winger Nathan Redmond knocked back.