The defender claimed he had not been given a reason for being frozen out at the Foxes.
The 30-year-old, who has a deal until 2024, told Danish publication Ekstra Bladet this month his first team exile was 'strange'.
Vestergaard has not started a Premier League game since January 2022 and had the chance to join Fulham last year, also turning down a move to Hetha Berlin in January as his partner was heavily pregnant.
Rodgers declined to comment on suggestions Vestergaard, who joined from Southampton in 2021, had been banished from the first team squad but has spoken to him.
Rodgers said: "I've not talked too much on it because it was an unauthorised interview. I've spoken to Jannik and I had to make sure I was talking to him as I'd seen in his quote he didn't know why (he was not playing).
"I had to make sure I wasn't talking to his twin brother for the last 14-15 months.
"We have respected him when he has decided to stay and that communication has always been there. The communication with him and the club has always been there, about where he stands.
"I was surprised when I saw it. Anything which has happened we will deal with internally and we'll move on from there.
"Any success we've ever had has been with solidarity on and off the pitch and that's something I'll always look to protect."
The Foxes host Chelsea on Saturday two points above the Premier League relegation zone after last week's 1-0 defeat at Southampton.
Four straight defeats in all competitions has seen them slip closer towards the bottom three and suffer a shock exit to Blackburn in the FA Cup.
Chelsea arrive having reached the last eight of the Champions League after knocking out Borussia Dortmund 2-1 on aggregate on Wednesday.
It eased the pressure on manager Graham Potter and Rodgers believes he deserves time at Stamford Bridge.
He said: "It's part of the job at this level. When you have experience you understand it's there. I've always accepted that as part of the job. if you are not winning games you are under pressure.
"Graham is very new into them, it's a difficult challenge he has because of him having so many players. He is an excellent coach, it's just time. The club has gone through a lot of change.
"The trouble is when you are at the top you may not get the time but he is very much deserving of it."
Harvey Barnes may return on Saturday after an ankle problem and Jonny Evans (thigh) could also make the squad while Youri Tielemans (ankle), Victor Kristiansen (ankle) and James Justin (Achilles) remain out.