The Foxes lost 6-2 to Antonio Conte's side at the Tottenham Hotspurs Stadium to leave them bottom of the Premier League table.
Leicester are the only side without a win in the top flight following a run of six straight defeats.
Rodgers has been in charge of the Foxes since February 2019 and led them to their first FA Cup triumph last year but the Northern Irishman knows his position has come under scrutiny after a poor start to the season.
"I come in every day and do my work. I totally understand the frustration of supporters. I can't hide from that. It's my responsibility," he told Sky Sports.
"Whatever happens I'll have a huge amount of respect for [the owners] because they've given me great support. I understand the game.
"The scoreline didn't reflect the game but the bottom line is it's a heavy defeat.
"They've given me brilliant support. Whatever happens to me at Leicester, whether I stay and fight on, I'll always respect them."
Meanwhile, midfielder James Maddison believes the club's poor form should not be attributed to Rodgers alone as the players also have to take responsibility.
"It is tough to debrief it in my head so quickly after," he said. "People will check the scoreline and see 6-2 and think 'wow, Leicester got battered again' but I don't think that was the case.
"We've worked so hard on the training ground this week because we know we need to produce better results and in the first-half it felt like the Leicester we know and the Brendan Rodgers side we've produced in the last few years, with the high press and creating chances. It is tough to take.
"We conceded from a couple of a set-pieces which is always disappointing, especially if I told you how many hours we've spent on that because it was a weakness last year.
"We've brought in a set-piece coach and everyone has really bought into that.
"The players should definitely be taking some of the brunt. It is a collective, it's not the manager, we are an XI out there.
"It hurts when you really respect the manager, and ultimately we are the players out there and we haven't been delivering recently."
