Hojbjerg, 24, has repeatedly been linked with a move away from the club and recently expressed a desire to play in the Champions League.
The Denmark international, signed from Bayern Munich in 2016, has just over 12 months left on his contract.
England midfielder Ward-Prowse will take over as captain for the remainder of the 2019-20 season.
Speaking at a press conference prior to the resumption of the Premier League, Hasenhuttl said: “No, he [Hojbjerg] will not be captain.
“We had an agreement, and [I] made clear that if you decide not to extend your contract, you will not be captain of this team.
“He knows that he is an important player for this club. I think we have found a good agreement.”
Hojbjerg has made more 100 Premier League appearances for Southampton.