The Foxes have moved for Denmark international Vestergaard, 29, after Wesley Fofana suffered a fractured fibula during a friendly against Villarreal.
A £15million deal is reported to have been agreed, with Saints boss Ralph Hasenhuttl confirming the remaining medical details were being finalised.
Vestergaard would need to be registered by noon on Friday to feature against Wolves for the FA Community Shield winners.
“Clearly there has been contact by the clubs. There has been nothing finalised but hopefully we will be able to confirm that,” Rodgers said at a press conference on Thursday afternoon.
“We have permission from them for him to be here. Hopefully everything can be organised, then we will see if he is available for the weekend.
“Without going into detail before Jannik is confirmed as a signing, it was pretty clear we needed a centre-half. That’s probably what sparked the interest, we were very light in the area.
“It’s a great tribute to the club that we could get our work done very quickly and see if we could sign a player of quality. Once we have that done, I can speak more on it.”
Fofana had surgery this week and Rodgers revealed the Frenchman was staying positive as he faces up to a lengthy spell of rehabilitation.
“He’s doing very well, is coming back home and he’s in great spirits,” said the Foxes boss.
“It’s hugely disappointing for him and for us. He’s had the operation and it’s gone very well.
“Now it’s all about his recovery, but he’s in good spirits and we’ll look forward to seeing him when he can come in.”
Timothy Castagne continues to be assessed on the eye injury he sustained while with Belgium at Euro 2020.
Northern Ireland centre-back Jonny Evans is also still recovering from a foot problem as he targets a return following the international break.
On Thursday evening, Leicester confirmed winger Rachid Ghezzal has signed a permanent deal with Turkish Super Lig side Besiktas, subject to international clearance.
The 29-year-old joined the Foxes from Monaco during the summer of 2018 and also had loan spells at Fiorentina and Besiktas.
Earlier, Leicester revealed initial plans for the expansion and development of the King Power Stadium as part of a public consultation ahead of a planning submission, which is due to be made this autumn.
The plans will include proposals for the addition of 8,000 seats, taking the capacity to 40,000, as well as a 220-room hotel, also with a multi-purpose event and entertainment arena.
Proposals include new retail space for the club, residential and commercial space, along with multi-storey car parking facilities.
Rodgers said: “It’s an incredible tribute to (former owner) Khun Vichai (Srivaddhanaprabha) and his legacy.
“You see some clubs and the decline that can set in but the strength shown to continue this amazing journey, it’s been an incredible success.
“What I love about the ownership is their will to be better. They want to make the facilities better and that aligns with my view.
“The development of the stadium will bring us more supporters and extra revenue and extra jobs for the city. Hopefully over the next decade we can continue that success.”