A reported £15million deal has been agreed for Vestergaard, who was in the final year of his contract with the Saints having joined from Borussia Monchengladbach during July 2018.
“He (Vestergaard) is doing the medical check, and when that is done, then the deal is done,” Hasenhuttl said as he prepared for Saturday’s trip to Everton.
“He will go to Leicester, then it is for us to replace him and this is what we will do in the future.”
Hasenhuttl also saw last season’s top scorer Danny Ings join Aston Villa earlier in the transfer window.
However the Austrian accepts there is little the club can do when in such a situation.
“Yes, we lost a few very important players in our club which was an absolute must because two of them only had one year left on their contracts,” he said.
“For the club it’s the only way you can go because the rules in the transfer market are like this. When you are coming into the last two years of a contract with a player then the power moves to the player.
“You do not let them go without a transfer fee. That would be a disaster for every club.
“We had to do these deals and finally we are happy we made these deals and could replace them with new players, made the squad bigger with players with potential to develop.”
Hasenhuttl added: “Overall it must be the sporting success and this is always the biggest challenge for a manager.
“You lose your best scorer and still there is demand to have the most possible success, but this is our job and I am not the only manager who loses the best players and then needs to perform well.”
Following Ings’ departure, Southampton moved quickly to bring in forward cover by signing Adam Armstrong from Blackburn, while Armando Broja, 19, also arrived on a season loan from Chelsea.
Teenage defender Valentino Livramento is another permanent summer signing from the Blues, while left-back Romain Perraud was the first to join at the start of the transfer window.
After Vestergaard’s sale, Hasenhuttl expects to be able to bring in more defensive reinforcements, but otherwise feels the squad is well set for the challenges ahead.
“We have lost Jannik now, so in the centre-back position we are not super strong with the numbers, but the rest of the squad seems to be quite balanced,” the Saints boss said.
“We have enough experience in the team, but it is more important that our players work hard and put everything on the pitch that we need to do.”
Midfielder James Ward-Prowse has returned to training from a knee problem, and is expected to be available to face Everton.
Hasenhuttl added: “Now we have a wider squad, a bigger squad and in the end a lot of games are coming up, so everyone must bring themselves in the best physical and mental condition to help us.”