The 28-year-old's current three-year deal expires in the summer of 2022 and he has yet to commit to extending his stay with the Saints amid reported interest from Tottenham.
Hasenhuttl said: "As long as there is no signing of the contract, a new one, (the question) will not stop I think.
"I can only say that we want him to stay here. We've shown as a club that we want him to be here, definitely, and he's shown signals that he wants to stay with us. So I'm still positive that we're fine together for the future."
"For us, it's important to show him that we don't want to sell him, we don't want to sell him next summer, we want him to stay with us as long as possible," Hasenhuttl said.
"I think he enjoys what we're doing here. He knows his quality here. We helped him to be strong like he is at the moment, so it's obviously a win-win situation if we can find (a solution) together."
It has been reported that the Saints are ready to make Ings their highest-paid player, but the former Bournemouth, Burnley and Liverpool striker is still wrangling over the size of his release clause.
"There is no issue with him," Hasenhuttl added. "He is absolutely happy here I think. He likes to work with the team, he likes playing in this team.
"It is fantastic, a very good atmosphere here and there are some targets also for the season."
Ings missed six weeks after knee surgery in November and then sustained a hamstring injury in December before testing positive for Covid-19.
He has scored seven goals in 13 Premier League appearances, including the winner against former club Liverpool earlier this month.
Hasenhuttl made seven changes for Tuesday night's FA Cup win against Shrewsbury and is expected to field a blend of youth and experience again on Saturday.
Saints are scheduled to face Arsenal twice in three days, with the Gunners returning to St Mary's for a Premier League fixture next Tuesday.
When Aston Villa visit St Mary's in the league a week on Saturday, Hasenhuttl's side will be playing their fifth game in 15 days.