Liverpool manager Klopp rents the house on Merseyside that Rodgers lived in during his time in charge at Anfield.
The German joked on Thursday that he would withhold payment in an attempt to help motivate Leicester ahead of their game against Manchester City in 10 days’ time.
“I don’t think it’s a secret [that I’m his landlord]. It has been well-documented,” said Rodgers, whose team host Arsenal on Sunday before facing Pep Guardiola’s side.
“I saw what he said, he is a funny guy. He has been a very, very good tenant. I don’t need to keep an eye on him. He always pays up on time. He’s very good!
“We have a game before that but my job is to do Leicester a favour. Every game is important. These are exciting games in this last period of the season and we will look to give our best and finish the season as strongly as we can.”
Liverpool can return to the top of the Premier League table on Friday night with victory against relegated Huddersfield at Anfield.
Rodgers praised both teams fighting for the title in a race that could see the second and third-highest top-flight points totals in history.
“A lot can happen in these last few games—I only know too well about that,” said Rodgers, who came close to winning the Premier League with Liverpool in 2014.
“There are three games to go and it was at that stage it sort of disappeared for us, but both teams have been absolutely brilliant this season.
“I thought Manchester City were great the other night in dealing with the occasion of the derby game and in particular the second half, the level of their game was fantastic.
“Liverpool have been in the hunt, seven points clear earlier on in the season, but they’ve fought really well and will continue to do so.”
On his own club’s game against Arsenal on Sunday, the Leicester boss said: “We’re looking forward to it and we’re excited about it.
“We’re looking forward to bringing our own intensity into the game and making it really difficult for Arsenal. We want to finish the season as strongly as we can.”