The former Liverpool manager, 46, had previously dismissed speculation linking him to the Arsenal job.
“Having that ownership here gives you great confidence that you can progress together,” he said.
“I hope we can continue to break records, win some trophies, and ideally be in a healthy position in the table.”
Rodgers’ new contract builds on the initial deal until 2022 that he signed when joining the Foxes from Celtic in February.
The Northern Irishman said the Leicester job is a chance to defy football’s financial restrictions and upset the Premier League’s established powerhouses.
“We’re a wealthy club with fantastic owners, and yet we don’t have the finances of a top-six club, so it’s about whether we can do it another way, a different way” he said.
Leicester have won 17 of their 26 Premier League matches under Rodgers and are second in the table -but the Foxes boss expects his side to get even better in 2020.
“There’s a busy period with a lot of points to play for, and then we’ve the second half of the season to look forward to, which is when my teams normally improve,” he said.
“That’s because they understand my methods even more, so hopefully we can do that this time.”
Leicester’s next Premier League game is away at Aston Villa on Sunday…Read More